Also known as “integral calculus.” A fancy word for finding the antiderivative or integral of a function—the opposite of finding a derivative or slope—which is a more accurate process to find the area bounded by a curve than summing the area of rectangles and taking limits.

Although differentiation is taught before integration, however, historically, the ideas of integration were developed before those of differentiation—the discovery that differentiation and integration are opposite processes, as revealed by the Fundamental Theorem of Calculus, came much later.

via giphy

by MathPlus August 02, 2018

Don't listen to the guy above talking about integration. Integration is NOT the reversal of differentiation. That would be the anti-derivative. Integrals and anti-derivatives are NOT the same thing. But they are connected by the Fundamental Theorem of Calculus.

If a function f(x) has an anti-derivative F(x), the area under the curve from a to b is equal to F(b)-F(a).

This is integration defined.

This is integration defined.

by MIT 2010 January 14, 2007

the reverse process of differentiaton

we know that, for example if f(x) = 2x^3 - 5x^2 + 3x -7

then f'(x) = 6x^2 - 10x + 3

This process can be reversed.

In general, y = x^n -> dy/dx = nx^(n-1)

So, reversing this process, it would seem that dy/dx = x^m -> y = (1/(m+1))x^(m+1)

The general process of finding a function from its derivative is known as interation.

we know that, for example if f(x) = 2x^3 - 5x^2 + 3x -7

then f'(x) = 6x^2 - 10x + 3

This process can be reversed.

In general, y = x^n -> dy/dx = nx^(n-1)

So, reversing this process, it would seem that dy/dx = x^m -> y = (1/(m+1))x^(m+1)

The general process of finding a function from its derivative is known as interation.

Given that dy/dx = 12x^2 + 4x - 5, find an expression for y.

y = 12((x^3)/3) + 4((x^2)/2) - 5((x^1)/1)

It would seem that

y=4x^3 + 2x^2 - 5x

but that is not quite the complete answer

Whenever you differentiate a constant you get zero,

e.g. y = 7 dy/dx = 0

and so the expression for y above could have any constant on the end and still satisfy dy/dx = 12x^2 + 4x - 5

The answer to this example is therefore

y= 4x^3 + 2x^2 - 5x + c, where c is a constant.

y = 12((x^3)/3) + 4((x^2)/2) - 5((x^1)/1)

It would seem that

y=4x^3 + 2x^2 - 5x

but that is not quite the complete answer

Whenever you differentiate a constant you get zero,

e.g. y = 7 dy/dx = 0

and so the expression for y above could have any constant on the end and still satisfy dy/dx = 12x^2 + 4x - 5

The answer to this example is therefore

y= 4x^3 + 2x^2 - 5x + c, where c is a constant.

by hotgirl69xxx December 22, 2004

by Atticus Coon March 07, 2005

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