Integral

The Simple English Wiktionary has a definition for: integral.

In calculus, an integral is the space under a graph of an equation (sometimes said as "the area under a curve"). An integral is the reverse of a derivative, and integral calculus is the opposite of differential calculus. A derivative is the steepness (or "slope"), as the rate of change, of a curve. The word "integral" can also be used as an adjective meaning "related to integers".

The symbol for integration, in calculus, is: ${\displaystyle \textstyle \int _{\,}^{\,}}$ as a tall letter "S".^[1][2][3]

Integrals and derivatives are part of a branch of mathematics called calculus. The link between these two is very important, and is called the fundamental theorem of calculus.^[4] The theorem says that an integral can be reversed by a derivative, similar to how an addition can be reversed by a subtraction.

Integration helps when trying to multiply units into a problem. For example, if a problem with rate, ${\displaystyle \left({\tfrac {\text{distance}}{\text{time}}}\right)}$, needs an answer with just distance, one solution is to integrate with respect to time. This means multiplying in time to cancel the time in ${\displaystyle \left({\tfrac {\text{distance}}{\text{time}}}\right)\times {\text{time}}}$. This is done by adding small slices of the rate graph together. The slices are close to zero in width, but adding them together indefinitely makes them add up to a whole. This is called a Riemann sum.

Adding these slices together gives the equation that the first equation is the derivative of. Integrals are like a way to add many tiny things together by hand. It is like summation, which is adding ${\displaystyle 1+2+3+4....+n}$. The difference with integration is that we also have to add all the decimals and fractions in between.^[4]

Another time integration is helpful is when finding the volume of a solid. It can add two-dimensional (without width) slices of the solid together indefinitely—until there is a width. This means the object now has three dimensions: the original two and a width. This gives the volume of the three-dimensional object described.

1. ↑ "List of Calculus and Analysis Symbols". Math Vault. 2020-05-11. Retrieved 2020-09-18.
2. ↑ Weisstein, Eric W. "Integral". mathworld.wolfram.com. Retrieved 2020-09-18.
3. ↑ "Integral calculus - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2020-09-18.
4. ↑ ^{4.0 4.1} Barton, David; Stuart Laird (2003). "16". Delta Mathematics. Pearson Education. ISBN 0-582-54539-0.