Piecewise Function Integral Calculator

This piecewise function integral calculator evaluates the integral of functions defined differently on different intervals. It handles piecewise functions by breaking them into their component parts and integrating each part separately.

How to Use This Calculator

To calculate the integral of a piecewise function:

Enter the function definition in the text area, specifying different expressions for different intervals.
Enter the lower and upper limits of integration.
Click "Calculate" to compute the integral.
Review the result and the step-by-step calculation.

Example format for a piecewise function: x^2 for x < 1, x+1 for x >= 1

How Piecewise Integrals Work

Piecewise functions are defined differently on different intervals. To integrate them, we:

Identify the intervals where the function changes definition.
Integrate each part separately over its respective interval.
Sum the results of each individual integral.

For a piecewise function f(x) = {f₁(x) for a ≤ x < b, f₂(x) for b ≤ x ≤ c}, the integral is:

∫[a to c] f(x) dx = ∫[a to b] f₁(x) dx + ∫[b to c] f₂(x) dx

Worked Examples

Example 1: Simple Piecewise Function

Function: f(x) = {x for 0 ≤ x < 1, x² for 1 ≤ x ≤ 2}

Integral from 0 to 2:

∫[0 to 1] x dx + ∫[1 to 2] x² dx = [x²/2]₀¹ + [x³/3]₁² = (1/2 - 0) + (8/3 - 1/3) = 0.5 + 7/3 ≈ 2.6667

Example 2: Absolute Value Function

Function: f(x) = {x for x ≥ 0, -x for x < 0}

Integral from -1 to 1:

∫[-1 to 0] -x dx + ∫[0 to 1] x dx = [x²/2]₋₁⁰ + [x²/2]₀¹ = (0 - 1/2) + (1/2 - 0) = -0.5 + 0.5 = 0

Frequently Asked Questions

Can this calculator handle functions with more than two pieces?

Yes, you can define as many pieces as needed in the function definition, specifying the intervals for each piece.

What if the function is not continuous at the boundary?

The calculator will still compute the integral, but you should verify the result matches your expectations for the specific function.

Can I use trigonometric or logarithmic functions?

Yes, you can include trigonometric (sin, cos, tan) and logarithmic (ln, log) functions in your piecewise definition.