Piecewise Function Integral Calculator

Calculator Form

Example Data Table

Formula Used

This calculator evaluates a definite integral over a piecewise function by splitting the chosen bounds into the parts that overlap each defined piece.

For each active overlap, it applies:

∫_a^b f(x) dx = Σ ∫_Li^R_i f_i(x) dx

Here, L_i = max(a, piece start) and R_i = min(b, piece end).

Each segment integral is approximated numerically with the composite Simpson rule:

∫_L^R f(x) dx ≈ h/3 [f(x₀) + f(x_n) + 4Σf(x_odd) + 2Σf(x_even)]

where h = (R − L) / n and n is an even number of steps.

Gaps between defined pieces are treated as zero on the uncovered parts of the chosen interval.

How to Use This Calculator

1. Enter the lower and upper integration bounds.
2. Choose an even Simpson step count for accuracy.
3. Select how many piece rows you want to use.
4. Fill each row with interval start, interval end, and the matching expression.
5. Use non-overlapping intervals for a valid piecewise definition.
6. Click the calculate button to view the summary above the form.
7. Review the contribution table, notes, and Plotly graph.
8. Download the result as CSV or PDF when needed.

FAQs

1. What does this calculator compute?

It computes the definite integral of a piecewise function over chosen bounds. It also shows segment contributions, absolute area, mean value, and a graph.

2. Can the selected bounds cross multiple pieces?

Yes. The calculator automatically splits the interval wherever your bounds intersect defined pieces, then adds those segment integrals together.

3. What happens if there is a gap between pieces?

Any uncovered subinterval is treated as zero. The notes section lists those uncovered intervals so you can check your piecewise definition.

4. Are overlapping intervals allowed?

No. Overlapping pieces are rejected because a standard piecewise function should assign one rule to each x-value in the interval.

5. Does the order of bounds matter?

Yes. Reversing the bounds changes the sign of the definite integral. The calculator reports that in the notes section.

6. Which expression formats are supported?

You can use x, parentheses, +, -, *, /, ^, sin, cos, tan, sqrt, abs, exp, ln, log, pow, pi, and e.

7. Why use more Simpson steps?

More steps usually improve numerical accuracy, especially for curved or rapidly changing pieces. The step count must stay even for Simpson integration.

8. What do the CSV and PDF downloads contain?

The exports include the main summary values and the piece contribution table, making the result easy to store, print, or share.