Integral of A Piecewise Function Calculator

This calculator computes the integral of piecewise functions by evaluating each segment separately and summing the results. Piecewise functions are defined by different expressions over different intervals, and their integrals require careful evaluation at each interval boundary.

How to Use This Calculator

To calculate the integral of a piecewise function:

Enter the function definition in the "Function" field using interval notation (e.g., "x^2 for x ≤ 0, x for x > 0").
Specify the lower and upper limits of integration.
Click "Calculate" to compute the integral.
Review the result and chart visualization.

Note: The calculator assumes the function is continuous at the interval boundaries. For discontinuous functions, additional care is needed.

Formula for Piecewise Integrals

The integral of a piecewise function is calculated by integrating each segment separately and summing the results. For a function defined as:

f(x) = {
  f₁(x) for a ≤ x ≤ b,
  f₂(x) for b < x ≤ c,
  ...
}

The integral from a to c is:

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

Worked Example

Consider the piecewise function:

f(x) = {
x² for -1 ≤ x ≤ 0,
x for 0 < x ≤ 1
}

The integral from -1 to 1 is calculated as:

∫[-1,1] f(x) dx = ∫[-1,0] x² dx + ∫[0,1] x dx

Which evaluates to:

= [x³/3] from -1 to 0 + [x²/2] from 0 to 1
= (0 - (-1/3)) + (1/2 - 0)
= 1/3 + 1/2 = 5/6

Frequently Asked Questions

What if the function is discontinuous at an interval boundary?: The calculator assumes continuity. For discontinuous functions, you must handle the discontinuity separately by evaluating the limit from both sides.
Can I use trigonometric or logarithmic functions?: Yes, the calculator supports basic mathematical functions including trigonometric and logarithmic functions.
How accurate are the results?: The calculator uses numerical integration methods for complex functions, providing accurate results within standard computational limits.
Is there a limit to the number of intervals I can define?: The calculator can handle up to 10 intervals, which is sufficient for most practical applications.