Definite Integral Piecewise Function Calculator

How to Use This Calculator

To calculate the definite integral of a piecewise function:

1. Enter the function definition in the text area. Use semicolons to separate different cases. For example: "x^2; 3x+1; 5" for a function defined as x² for x ≤ 0, 3x+1 for 0 < x ≤ 1, and 5 for x > 1.
2. Specify the lower and upper bounds of the integral.
3. Click "Calculate" to compute the result.

The calculator will evaluate the integral by breaking it into parts where the function definition changes and summing the results.

Formula Explained

The definite integral of a piecewise function f(x) from a to b is calculated by:

For a piecewise function defined as:

f(x) = { f₁(x) for a ≤ x ≤ x₁, f₂(x) for x₁ < x ≤ x₂, ..., fₙ(x) for xₙ < x ≤ b }

The integral becomes the sum of integrals over each interval where the function definition is continuous.

Worked Examples

Example 1: Simple Piecewise Function

Calculate ∫[0,2] f(x) dx where f(x) = { x for 0 ≤ x ≤ 1, 2 for 1 < x ≤ 2 }

The integral is calculated as:

∫[0,1] x dx + ∫[1,2] 2 dx = [x²/2]₀¹ + [2x]₁² = (1/2 - 0) + (4 - 2) = 2.5

Example 2: More Complex Piecewise Function

Calculate ∫[-1,3] f(x) dx where f(x) = { x² for -1 ≤ x ≤ 0, x+1 for 0 < x ≤ 1, 2 for 1 < x ≤ 3 }

The integral is calculated as:

∫[-1,0] x² dx + ∫[0,1] (x+1) dx + ∫[1,3] 2 dx = [x³/3]₋₁⁰ + [(x+1)²/2]₀¹ + [2x]₁³ = (0 - (-1/3)) + (2 - 0.5) + (6 - 2) = 1/3 + 1.5 + 4 = 6.833...