Definite Integration of Absolute Value Functions
How to solve definite integration of absolute value functions problems: example and its solution.
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