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To discover a fuss free guide to understanding integrals and using calculations which feature improper integrals, simply continue reading. As a bonus, you’ll also get to understand the major differences between improper integrals and their opposite, which are simply referred to as proper integrals.

What is an improper integral?

Whether you’re confused about what an improper integral is or you’re never come across the term improper integral, it may be helpful to be provided with a clear definition of an improper integral.

An improper integral, features a denominator which as a larger degree, then the degree of its numerator. Furthermore, an improper integral is a type of definite integral which has two infinite limits or which features an integrand which reaches infinity at one or more points in a defined range of integration.

It’s also worth keeping in mind that you can’t use a Riemann integral with improper integrals, like you can with other integrals.

What is an integrand?

If you’re unsure of what the term integrand refers to, in mathematics integrand refers to a function, which you will then integrate.

How you can use improper integrals in your calculations and equations:

One of the most common equations which features improper integrals, seeks to convert improper integrals into mixed numbers.

How to perform calculations which feature improper integrals:

While it’s certainly possible to perform calculations which involve the use of improper integrals manually, it’s far simpler to use an improper integral calculator to get quick answers and solutions to your problems.

As an example, you’ll easily be able to turn any improper integral into mixed numbers, in less than a minute, which is incredibly convenient and well worth experimenting with. Especially if you forsee yourself performing such calculations on a regular basis as if you use a calculator to turn improper integrals into mixed numbers, you’ll save yourself a lot of time and energy in the long run!

What is the opposite of an improper integral?

The opposite of an improper integral is simply referred to as a proper integral.

What is a proper integral?

A proper integral features a denominator degree which is less than the degree of its numerator. Remember to keep in mind that in mathematics the term denominator refers to the bottom figure in a fraction, while the term numerator is used to refer to the top number in a fraction.

How does a proper integral differ from an improper integral:

If you’re a bit perplexed by how a proper integral differs from an improper integral, while a proper integral has a denominator degree which is less than the degree of its numerator an improper integral has a denominator degree which is larger than the degree of its numerator.

Hopefully you now have a clear understanding of the use of the both improper and proper integrals and know how to go about using a calculator to solve any equations which involve improper integrals. If you ever forget the difference between improper integrals and proper integrals simply refer back to this handy page.

To discover everything you need to know about the element Barium, simply continue reading to discover a handy guide to Barium.

Everything you need to know about Barium:

Classification:
Is Barium a metal nonmetal or metaloid? Barium is classed as an Earth metal.

Atomic number:

Barium has the atomic number 56.

Symbol:

Barium’s atomic symbol is BA.

Appearance:

Barium is silver in color and is a soft, pliable metal. Barium has a tendency to tarnish quickly when it is exposed to air and also is known to react to water. When Barium is ultra pure it sometimes has a slightly golden tinge.

Typical uses:

Barium is sometimes used in drilling fluids, that businesses use that operate in the mining and oil industries. Less commonly Barium is also sometimes used as an ingredient in paint.

Barium is also used in the health care industry. A type of non toxic Barium, Barium Sulfate is occasionally prescribed to patients who complain of serious digestive issues.

Once ingested Barium Sulfate is processed by the patients’ stomach and intestines and shows up on x-rays and allows doctors to analyse a patients stomach and intestines as Barium Sulfate is a heavy element and shows up well in x-rays. You may be surprised to read that Barium has been used in x-rays since 1908.

Is Barium toxic?

Yes, Barium is toxic and all Barium compounds are classed as being toxic. Which is one reason why Barium is not used as commonly as other metals. However, Barium Sulfate which was mentioned above is not toxic and can be safely consumed.Barium Ion is often used in pesticide products as its so toxic.

How toxic is Barium to humans? Barium is extremely poisonous and even one gram of Barium can be lethal if its ingested by humans. Individuals who accidentally consume Barium are instructed to drink water in order to flush the barium which they have ingested out of their system.

Does Barium have a known biological role?

No, Barium doesn’t have a clear, known biological purpose. However, Barium Sulfate can be found in one species of algae.

Is Barium ever found with other compounds?

Yes, in fact Barium is only found with other compounds. As examples Barium combined with Barite is called Barium Sulfate and Barium which is combined with Whiterite is known as Barium Carbonate.

How is Barium created?

There are two known ways to produce Barium. The first way to produce Barium involves performing electrolysis on Molten Barium Chloride, while the second method of producing Barium entails heating Barium Oxide with Aluminium Powder.

When was Barium first discovered and who is attributed with discovering Barium?

Barium was first discovered by the scientist Humphrey Davy in 1808. When Davy managed to isolate Barium by utilizing electrolysis. Another scientist Carl Scheele identified Barium in 1774 but Scheele is not credited with discovering Barium as he was unable to succesfully seperate and isolate Barium.

How did Barium get its name?

Barium stems from the Greek word barys which roughly translates to the English word heavy.

Hopefully you now have a clear understanding of the properties, uses and dangers of the solid metal element Barium.

If you’re confused about whether negative numbers and decimal figures can be rational numbers, continue reading to demystify both types of numbers!

What is the key difference between rational numbers and their irrational counterparts?

Before you’re able to figure out whether a given number is a rational number or an irrational number, it’s important to be able to tell the key difference between both types of numbers.

Rational numbers are figures which can be easily written down as a whole fraction. Whereas their opposite, irrational numbers can’t be written down as a simple whole fraction.

You may be surprised to read that both decimal figures and negative figures can be classed as irrational numbers as well as rational numbers. Remember that if a negative number or a decimal number can be written down as a whole fraction, you’re dealing with a regular figure.

Are negative numbers rational?

As mentioned briefly above negative numbers can be rational or irrational, to figure out whether your negative number is rational, go ahead and try and turn your negative number into a whole fraction.

Another tip which you may want to use in order to tell if the number that you’re looking at is rational or irrational, check whether your negative number’s positive equivalent is a rational number. As if a negative number’s positive counterpart is negative, there’s a high likelihood that the number that the negative number that you’re trying to place is also an irrational number.

Is negative 1.5 a rational number?

If you want to have a go at classifying a negative number as a rational number or an irrational number, you may want to start off with an easy example, that is simple to sort such as -1.5. If you’re stuck, just make sure to try and write down -1.5 as a whole fraction. If you can successfully write down -1.5 as a whole fraction which has a whole numerator and a whole denominator, then -1.5 is a rational number. However, if you can’t turn -1.5 into a whole fraction, then -1.5 is an irrational number, not a rational number.

Examples of negative irrational numbers:
If you find it helpful to look at a list of examples of negative irrational numbers, in order to more easily be able to classify negative numbers, the following numbers are all negative irrational numbers, -2, -13 and -8.
Examples of decimal numbers that are irrational:

Just as negative numbers can be irrational, decimal numbers can be irrational too. One prime example that is great to keep in mind is the number pi. Pi is an infinite number, which doesn’t have reoccurring digits and which therefore is an irrational number. As there is no logical way that you can turn an infinite, endless number which has no end into a neat whole fraction.

Just be careful when you’re classing numbers not to automatically assume that decimals are always irrational as decimals which recurring numbers are almost always rational numbers.

If you’re ever confused about negative numbers or decimal numbers and whether they are rational or not, simply refer back to this handy article for answers.

If you’re interested in studying statistics and are interested in learning about the definitions of key statistical terms such as least to greatest, mean, mode, modal and median, simply continue reading to discover a guide to commonly used statistical terms. That will help you solve simple and complex statistical equations with ease.

Least to greatest definition:

In statistics data is always always arranged from least to greatest, which will always allow you to easily find the mean, median and mode of data.

The process of rearranging data from least to greatest is referred to simply as ordering. So if you’re ever asked to order your data while completing a statistical exercise, simply order your data from least to greatest.

What does the term mean refer to?

In the study of statistics, the term mean refers to the answer which you’ll get if you add all the items in your list and then divide your answer by the number of items in your list.

What does the term mode refer to?

The term mode refers to answer which you’ll get if you look at the items in your list and choose the value which appears in the highest frequency. Another word for mode is the modal value, so don’t feel thrown off if you see the term modal referred to in a statistics question.

If it’s not easy to find an answer by simply looking at your list of numbers, you can make things a lot easier for yourself by creating a simply tally chart.

When you create a tally chart, make sure that you have a unique column for each different value which is represented in your list. Then simply add a mark in each column each time that a specific value occurs in your list, to find your mode.

What does the term median refer to?

In statistics the term median refers to the middle number, in a series of numbers that has been sorted from least to greatest. If there are two numbers in the middle of your list, simply find the average of your two numbers, which can be done by adding both of your numbers together and then dividing your total by two.

What does the term range refer to in statistics?

It may be useful to remember that your list of numbers which range from least to greatest is always referred to as your range. Making sure to recall statistical terms such as range, mode, median, mean is crucial to be able to understanding statistical problems which you’re given.

As half the difficulty of being able to complete complex problems is being able to clearly understand what each problem is asking you to do. As if you understand each key term, you’ll know what functions and operations to use to solve your problems.

In conclusion:

Hopefully understanding the satistics terms which are listed above which include least to greatest, mean, median, modal, range and mode, will help you better understand the statistics equations which you’ll come across in your statistics studies!

The Lewis base list is commonly referred to in the study of chemistry and contains substances which donate an electron pair which is then shared by the base and the covalent bond. To learn more about some of the bases, which are contained in the Lewis base list, simply continue reading to discover everything you need to know about the Lewis base list.

Everything you need to know about the Lewis bases:

1. Most types of oxygen and nitrogen can act as Lewis bases
If a type of oxygen or nitrogen feature organic compounds, the oxygen or nitrogen type in question can be used as a Lewis base as it will feature a pair of electrons.
2. Atoms, ions and molecules can all act as Lewis bases
You may be surprised to read that atoms, ions and molecules can all be used as Lewis bases.
3. For an atom, ion or molecule to be used as a Lewis base, there also needs to be an atom which will accept a Lewis bases’ pair of electrons
If an atom, ion or molecule isn’t able to donate its pair of electrons to another atom, which will accept them, the atom, ion or molecule in question won’t become a Lewis base.
4. You can find out whether or not an atom, ion or molecule is by drawing the Lewis structure
If you’re curious about how you can accurately discover whether a particular atom, ion or molecule is a Lewis base, you can draw the Lewis structure in order to see whether you can find a pair of electrons which can be used to form a bond to another atom. If you can identify a pair of electrons which can be donated to another atom, the atom, ion or molecule which you’re researching can in fact act as a Lewis bond.
As an example, ammonia may be able to be used in a Lewis base if it’s able to donate a pair of electrons to the element H+, in order to form a bond with the element H+.
5. In organic chemistry compounds with pi electrons are usually Lewis bases
If you study organic chemistry, you’ll discover that the vast majority of compounds which boast pi electrons can be classified as Lewis bases.
6. Some compounds can be used either as a Lewis base or a Lewis acid
To further complicate things, some compounds, including atoms, ions and molecules actually have the potential to turn into either a Lewis base or a Lewis acid. As an example, most forms of alcohol act as acids when they donate H+ but can also act as a Lewis base when their oxygen atom decides to accept H+.
So if you’re interested in learning further information about Lewis bases, it’s well worth researching further types of compounds which can be turned into Lewis bases and Lewis acids.
So if you were confused about how compounds such as ions, molecules and atoms can be used as Lewis bases, hopefully you now have a clear understanding of Lewis bases.

In order to discover how you can use the radial acceleration formula in order to perform complex calculations about items which move in a circular motion, simply continue reading to discover a guide to radial acceleration.

What does the term radial acceleration refer to?

Whether you’ve heard the term radial acceleration before and were confused about its definition or you’ve just come across the term radial acceleration, you may be curious about the definition of radial acceleration.

Simply put, radial acceleration refers to the speed of any moving object, which has been altered, sometimes by the moving object rapidly changing direction. Radial acceleration typically measures acceleration of a moving object, which moves in a circular motion or pattern.

What are the units of measurement that are used to measure radial acceleration?

The units of measurement, which you’ll need to use in order to be able to measure radial acceleration include radians per second and meters per second squared.

What is a simple definition for the term centripetal acceleration?

The term centripetal acceleration is also referred to sometimes as radial acceleration and describes a moving object which continues to move in a circle and which has an acceleration which is directed to the middle of its circle.

What are the units of measurement that are used to measure centripetal acceleration?

The units of measurement that you would use in order to accurately measure centripetal acceleration include meters per second squared. Which happens to be the most used unit of measure to measure centripetal acceleration.

How to calculate the radial acceleration:

In order to calculate the radial acceleration, you’ll need to use the radial acceleration formula, to get an accurate answer. The radial acceleration formula involves dividing the velocity by the given radius.

Why does radial acceleration occur:

If you were wondering why radial acceleration occurs, the simple answer is that radial acceleration occurs when there is a change in the direction of the velocity.

What types of situations would you use a radial acceleration formula in?

Any time that you seek to measure the acceleration of an item which is moving in a circular pattern such as a merry go round, you can’t go wrong using the formula velocity divided by the given radius.

Another key example of a moving object that continuously moves in a circular pattern is planet Earth, which moves in a radial pattern around the sun and which can also be described as being directed by the centrifugal force, which in this case is the sun.

The Earth moving around the sun as an example of radial acceleration:

If you would like to measure how fast the Earth moves around the sun, you could use the acceleration formula which you’ve read about above, to find out. Assuming of course that you’d have all the numbers which you’d need to complete your calculation.

If you’re curious, the radius of planet Earth’s continuous orbit is 149.6 million kms and the Earth’s current rate of velocity is approximately 29800 meters per second. So the answer which you should get to your maths problem is 0.00593 meters per second squared.

Hopefully, you now have all the information which you need in order to be able to solve complex radial acceleration problems on your own.

Why might you need to rationalize a denominator? In mathematics the term denominator refers to the bottom number in a fraction. You may be wondering in what circumstances may you need to look at rationalizing a denominator, the bottom number in the fraction which you’re working with.

The answer is that if you ever discover a radical expression in your denominator, you’ll need to get to work on rationalizing your denominator.

Do you need to rationalize your numerator if it contains a radical expression?

If you’re curious about whether you need to rationalize your fraction if you discover a radical expression in the top number of your fraction, which is your numerator, your answer is no. There is absolutely no reason to remove a radical expression from your fraction.

How to rationalize your denominator:

To rationalize the denominator you’ll need to multiply your fraction by either a single term or a set of terms which will be able to remove the radical expression in your denominator, which you’re intent on getting rid of. If you believe that you’d find step by step instructions on removing radial expressions from denominators, simply follow the instructions below.

1. Identify the radical expression

Your first step should be to identify the radical expression in your denominator, which you’ll then work on eliminating from your fraction. What is a common example of a radical expression that you may find in one of your denominators? An example of a radical expression, that you may find in one of your denominators is a square root.

2. Find a suitable number to multiply both your numerator and your denominator with

Once you’ve identified your radical expression, such as a square root, it’s time to try and identify a number which you can successfully multiply your top number and your bottom number of your fraction with.

3. Now multiply the numerator and denominator by the radical which you would like to remove from your fraction

If you happen to be rationalizing a fraction which features a monomial, ensure that you your numerator and your denominator are both multiplied by the same number. As what you’re doing is multiplying your factors by 1.

4. Simplify your fraction

Now that you’ve completed the 3 steps listed above, it’s the right time to simplify your fraction.

How to rationalize a binomial denominator:

If you’re trying to remove a radical expression from a binomial denominator instead of a monomial denominator, simply follow the instructions listed below, in order to discover your answer.

1. Assess the components of your fraction

Start off by assessing the components of your fraction as if your fraction contains a sum of two terms in your denominator which are irrational, you won’t be able to multiply your fraction properly.

2 Multiple the conjugate of your denominator by your fraction

3. Simply your fraction

Rationalize the denominator calculator:

Alternatively, if you’d like to rationalize your fraction is a fraction of the time, it’s well worth using a calculator to solve your equation for you.

Hopefully you now have a clear understanding of how to go about rationalizing denominators in fractions, in order to get rid of radicals such as square roots.

If you’re interested in being able to solve two step equations without any issues, simply learn how to use the simple, fuss free, two step equation which is outlined below.

How to confidently solve two step equations with integers:

Step one:

Shift all your data that doesn’t have the variable, which you’re looking to the opposite side of your equation.

Step two:

Remember that whatever you do to one side of your equation, you’ll need to do to the opposite to the other side of your equation. As an example if you use a + side on the left side of your equation, you’ll need to place a – side on the right side of your equation.

An example of a simple two step equation:

If you’re still not confident about your ability to use a two step equation in order to be able to solve a simple equation, it may help to work through an example exercise. Let’s start off with a fairly simple equation, that is fairly easy to solve but will allow you to confidently test out your knowledge of both of the two steps which you read about above.

The test equation which we will be working with is 4x+3=7. To find the value of x, make sure to follow the directions which were explained in the previous paragraph.

Step one:

As you know your first step should be to shift all of your data that doesn’t have the variable which you’re looking for, to the opposite side of your equation. In our example you need to subtract three from both sides of your equation. You should have written down 4x+3-3=7-3, which should then give you 4x=4.

Step two:

Now you’ll need to go about diving each side of your formula by 4. The process which you should write down now is 4x/4=4/4 and x=1. If the answer which you worked out is different, simply go back to step one and double check all of the processes which you used, to make sure that you didn’t make a simple error, which you’ll easily be able to correct.

An extra example to work through:

In order to cement the information which you have just learned, it’s a great idea to work through one more example. This time the two step equation which you’ll need to solve is 3x to the power of 3+4=28. To find the value of x, simply work through the simple two step instructions, which you have been learning to follow.

Step one:

Again, begin by isolating the given variable in your equation. You should have written down 3x to the power of 4-4=28-4 and 3x to the power of 4=24.

Step two:

Lastly to find the value of x you’ll need to divide each side by 3. Your notes should now show 3x to the power of 3/3=24/3. X to the power of 3=8. Now write down the cube root for each side and you should get your answer, x=2.

Again, if you didn’t obtain the same answer, simply work through the two step by step instructions listed above again.

If you’re wondering what does corresponding mean in math, simply continue reading to discover a simple easy to understand answer.

What does corresponding mean in math?

Simply put the term corresponding in math refers to objects such as angles which appear in the same exact place, in two similar situations.

What are corresponding angles and how do you go about figuring out the value of corresponding angles?

The mathematical term corresponding angles is commonly used to refer to matching angles. What are corresponding angles? When two straight lines are crossed by a third line, which is known as a traversal line, the angles in each matching corner are referred to as corresponding angles. Corresponding angles are always the same size as each other.
As an example if one angle in a matching corner is 118 degrees, the angle is the corresponding matching corner of the second line will also be 118 degrees.
How to find corresponding angles?
If you label each angle from left to right and top to bottom, you can find corresponding angles by looking at the place of each angle. Angle a and angle e, will be corresponding angles. Angle b and angle f will be matching corresponding angles, angle c and angle g will also be corresponding angles and lastly angle d and angle h will also be even, corresponding angles.
What happens if two lines are crossed by parallel lines?
As a second example, if two straight lines, which do not touch are both crossed by two parallel lines, which each cross one of the two straight lines but don’t intersect each other, you’ll also be able to work out various corresponding angles.
In this example, again assuming that each angle is labelled from a to b, going left to right and top to bottom, angle a and angle e are corresponding angles, angle b and angle h are corresponding angles, angle c and angle f are corresponding angles and angle d and angle h are also corresponding angles.
While it may take a couple of minutes to understand the rules which govern corresponding angles, once you’re able to find out one corresponding angle, you should have no issues figuring out corresponding angles in the future.
Are there any other circumstances in maths in which the term corresponding is frequently used?
Yes, the term corresponding is often used to describe properties of congruent triangles. What is a congruent triangle? A congruent triangle is a common type of triangle where each side is exactly the same length as the other two angles. Congruent triangles also feature three internal angles a right angle and two congruent or equal angles.
As congruent triangles feature equal lines and angles, congruent triangles can also be said to feature corresponding lines and corresponding angles. If you’re ever trying to explain the term corresponding in maths to a friend, it’s a great idea to use this particular example, to illustrate the points which you want to make.
Hopefully you’re now confident about how the term corresponding is used in mathematics and how you can effortlessly figure out corresponding angles.

If you’re interested in learning about some of the features of regular polygons, simply continue reading to discover valuable information on a wide variety of polygons. Examples of which include a 10 sided polygon and a 100 sided polygon.

What is a 10 sided polygon called?

A 10 sided polygon is most commonly referred to as a decagon, although they are sometimes also referred to as a 10-gon.

What are the key features of a decagon?

A decagon is a regular polygon as it has 10 equal sides. All of a decagon’s internal angles each add up to 144 degrees. As a decagon’s internal angles are less than 180 degrees, a decagon can accurately be described as being a convex polygon.

What is a 9 sided polygon called?

A 9 sided polygon is called a nonogan.

What are the key features of a nonagon?

A nonagon has nine equal sides and as such is also a regular polygon. Just like a decagon a nonagon is also classed as a convex polygon as each of the congruent interior angles of a decagon measures 140 degrees and as mentioned above to be classified as an obtuse polygon a polygon needs to feature interior angles which measure at least 180 degrees.

What is an 8 sided polygon called?

An 8 sided polygon is referred to as an octagon. If you’ve ever watched a UFC, mixed martial arts event you’ll know that each fight takes place within an octagonal shaped ring which features 8 equal sides.

What are some of the key features of an octagon?

An octagon features 8 sides as well as 8 interior angles. Each interior angle inside an octagon measures exactly 135 degrees. An octagon is also a convex polygon.

What is a 7 sided polygon called?

A 7 sided polygon is most commonly referred to as a heptagon. However, a 7 sided polygon is also sometimes referred to as a septagon as “sept” is a latin prefix which is linked to the number 7. Whereas the term heptagon has Greek roots as the prefix “hepta” is connected with the number 7. If you’re curious the “agon” is also Greek and roughly translates to angle. So in Greek the term heptagon translates to 7 angles.

What are some of the key features of a heptagon?

By now you may have guessed that a heptagon has 7 congruent sides and 7 interior angles. Like all of the polygons listed above a heptagon is also a convex polygon as its interior angels each measure 123.57 degrees.

What is a 100 sided polygon called?

While we’ve been discussing polygons which have 10 or less sides, there are polygons which have a far greater number of edges. For starters, a regular polygon which boasts 100 edges is referred to simply as a hectogon. Although you can also accurately call a 100 sided polygon a hectatontogon.

What is a 10000 sided polygon called?

While it can be difficult to picture what a 10000 sided polygon would look like a 10000 sided polygon is referred to as myriagon. Like all of the polygons listed above a 10000 sided polygon is also a convex polygon.

Hopefully, you’re now intrigued enough to learn about more different types of polygons.