Absolute Value Integral Calculator

What is Absolute Value Integral?

An absolute value integral refers to the integration of a function that contains an absolute value expression. Absolute value functions are piecewise functions that output the non-negative value of their input. When integrating such functions, the integral must be split at the point where the expression inside the absolute value changes sign.

Absolute value integrals are commonly encountered in problems involving distance, work, and optimization. They are particularly useful when dealing with functions that change behavior based on their input value.

How to Calculate Absolute Value Integral

Calculating the integral of an absolute value function involves several steps:

1. Identify the point(s) where the expression inside the absolute value changes sign.
2. Split the integral into sub-intervals based on these points.
3. Remove the absolute value signs in each sub-interval by considering the sign of the expression.
4. Integrate each sub-interval separately.
5. Sum the results of the sub-interval integrals to get the final result.

This process ensures that the integral is calculated correctly by accounting for the piecewise nature of absolute value functions.

Formula

This formula provides the foundation for calculating absolute value integrals. The key is to properly handle the sign changes within the integral limits.

Example Calculation

Consider the integral ∫ |x - 2| dx from 0 to 4.

The function x - 2 changes sign at x = 2. Therefore, we split the integral into two parts:

1. ∫ |x - 2| dx from 0 to 2: Here, x - 2 is negative, so we use -∫ (x - 2) dx.
2. ∫ |x - 2| dx from 2 to 4: Here, x - 2 is positive, so we use ∫ (x - 2) dx.

Calculating each part:

• First part: -[(x²/2 - 2x)] from 0 to 2 = -[(2 - 4) - (0 - 0)] = 2
• Second part: [(x²/2 - 2x)] from 2 to 4 = [(8 - 8) - (2 - 4)] = 2

The total integral is 2 + 2 = 4.