Definite Integration of Absolute Value Functions

Definite Integration of Absolute Value Functions

How to solve definite integration of absolute value functions problems: example and its solution.

Example

Find the given integral. The integral, from -1 to 2, of |x^2 - 1| dx

Draw the graph y = |x2 - 1|.

In [-1, 1],
the graph of y = x2 - 1 (dashed)
is below the x-axis.

So draw the image of this dashed line
under the reflection in y = 0.
(which is above the x-axis.)

Graphing absolute value functions - Example 3

So y = |x2 - 1| is

-(x2 - 1) in [-1, 1],
x2 - 1 in [1, 2].

So change the given integral into two integrals:

-11 (-x2 + 1) dx + ∫12 (x2 - 1) dx.

See the first integral.

The interval of the integral is symmetric: [-1, 1].
And the terms in (-x2 + 1) are all even functions.

So ∫-11 (-x2 + 1) dx = 2 ∫01 (-x2 + 1) dx

Definite integration of odd and even functions

Solve the definite integrals.

Definite integration of polynomials