# Area between Two Functions

How to find the area between two functions: formula, proof, examples, and their solutions.

## Formula

The area between two functions, *f*(*x*) and *g*(*x*),

is the integral of |*f*(*x*) - *g*(*x*)| *dx*.*f*(*x*) is the upper function.

And *g*(*x*) is the lower function.

## Proof

See the sliced rectangle.

Its height is |*f*(*x*) - *g*(*x*)|:

the difference between two functions.

And its base is *dx*.

So the area of the sliced rectange is |*f*(*x*) - *g*(*x*)| *dx*.

So the area between two functions

is the integral of |*f*(*x*) - *g*(*x*)| *dx*.

## Example 1

First find the bounded region.*y* = *x*^{2} - *x*

= *x*(*x* - 2)

So the zeros of *y* = *x*^{2} - *x* are 0 and 1.

Quadratic functions - factored form

And *y* = *x*^{3} starts from 0 and goes upward,

faster than *y* = *x*^{2} - *x*.

So the gray region is the bounded region.

*y* = *x*^{3} is the upper function.*y* = *x*^{2} - *x* is the lower function.

And the integral interval is [0, 2].

So *A* = ∫_{0}^{2} [*x*^{3} - (*x*^{2} - *x*)] *dx*.

Solve the integral.

Definite integration of polynomials

To make the denominators the same,

multiply 3/3 by 6.

## Example 2

First find the bounded region.*y* = *x*^{2} - 3*x*

= *x*(*x* - 3)

So the zeros of *y* = *x*^{2} - 3*x* are 0 and 3.

Quadratic functions - factored form

And *y* = *x* passes through *y* = *x*^{2} - 3*x*

at *x* = 0 and the other intersecting point,

which is the upper limit of the integral.

So find the other intersecting point

by setting *x*^{2} - 3*x* = *x*.

Solving quadratic-linear systems

Then *x* = 0, 4.

So *x* = 4 is the other intersecting point.

*y* = *x* is the upper function.*y* = *x*^{2} - 3*x* is the lower function.

And the integral interval is [0, 4].

So *A* = ∫_{0}^{4} [*x* - (*x*^{2} - 3*x*)] *dx*.

Solve the integral.

Definite integration of polynomials