Area between Two Functions

Area between Two Functions

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

Formula

(area between two functions) = [inteval, from a to b, of |f(x) - g(x)| dx]

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

Area between Two Functions: Proof of the Formula

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

Find the area bounded by y = x^3, y = x^2 - x, and x = 2.

First find the bounded region.

y = x2 - x
= x(x - 2)

So the zeros of y = x2 - x are 0 and 1.

Quadratic functions - factored form

And y = x3 starts from 0 and goes upward,
faster than y = x2 - x.

So the gray region is the bounded region.

y = x3 is the upper function.
y = x2 - x is the lower function.
And the integral interval is [0, 2].

So A = ∫02 [x3 - (x2 - x)] dx.

Solve the integral.

Definite integration of polynomials

To make the denominators the same,
multiply 3/3 by 6.

Example 2

Find the area bounded by y = x^2 - 3x and y = x.

First find the bounded region.

y = x2 - 3x
= x(x - 3)

So the zeros of y = x2 - 3x are 0 and 3.

Quadratic functions - factored form

And y = x passes through y = x2 - 3x
at x = 0 and the other intersecting point,
which is the upper limit of the integral.

So find the other intersecting point
by setting x2 - 3x = x.

Solving quadratic-linear systems

Then x = 0, 4.

So x = 4 is the other intersecting point.

y = x is the upper function.
y = x2 - 3x is the lower function.
And the integral interval is [0, 4].

So A = ∫04 [x - (x2 - 3x)] dx.

Solve the integral.

Definite integration of polynomials