Integral Calculator Absolute Value

Calculating the integral of an absolute value function requires careful consideration of the function's behavior across its domain. This guide explains how to compute integrals involving absolute value functions, including both definite and indefinite integrals, with practical examples and a dedicated calculator.

What is Integral of Absolute Value?

The integral of an absolute value function is the area under the curve of the absolute value function between specified limits. Absolute value functions change their behavior at points where the expression inside the absolute value changes sign, which affects how the integral is computed.

For functions of the form |f(x)|, the integral must be split into intervals where f(x) is positive or negative. The integral of |f(x)| from a to b is equal to the integral of f(x) from a to b if f(x) is always positive on [a, b], or if f(x) is always negative, the integral is the negative of the integral of f(x).

How to Calculate Integral of Absolute Value

To calculate the integral of an absolute value function, follow these steps:

Identify the points where the expression inside the absolute value changes sign (i.e., where f(x) = 0).
Split the integral into subintervals where the expression inside the absolute value is always positive or always negative.
Compute the integral separately for each subinterval, removing the absolute value signs where appropriate.
Sum the results of the integrals over all subintervals to get the final result.

For definite integrals, ensure the limits of integration are correctly ordered and that the function is integrable on the interval.

Formula

The integral of |f(x)| from a to b is computed as:

∫[a,b] |f(x)| dx = ∫[a,c] f(x) dx + ∫[c,b] -f(x) dx, where c is the point where f(x) changes sign.

For indefinite integrals, the antiderivative of |f(x)| is piecewise defined based on the behavior of f(x).

Examples

Example 1: Definite Integral

Compute ∫[0,2] |x - 1| dx.

The function |x - 1| changes sign at x = 1. Splitting the integral:

∫[0,1] (1 - x) dx + ∫[1,2] (x - 1) dx = [x - x²/2] from 0 to 1 + [x²/2 - x] from 1 to 2 = (1 - 1/2) + (2 - 2 - 1/2 + 1) = 0.5 + 0.5 = 1.

Example 2: Indefinite Integral

Find ∫ |x - 1| dx.

The antiderivative is piecewise:

∫ |x - 1| dx = -1/2 (1 - x)² + C for x < 1, and 1/2 (x - 1)² + C for x > 1.

FAQ

What is the integral of |x|?

The integral of |x| is a piecewise function: -1/2 x² for x ≤ 0 and 1/2 x² for x ≥ 0. The definite integral from -a to a is a².

Can I integrate |sin(x)|?

Yes, but it requires splitting the integral at points where sin(x) changes sign, typically every π/2 radians.

How do I handle integrals of |f(x)| where f(x) has multiple sign changes?

Split the integral at each point where f(x) changes sign and compute the integral separately for each subinterval.