Integral of Absolute Value Calculator

Calculator input

Supported model: f(x) = ax³ + bx² + cx + d

a coefficient for x³

b coefficient for x²

c coefficient for x

d constant term

Lower bound

Upper bound

Decimal places

Graph points

Root scan intervals

Reset

Example data table

These worked examples show how the area changes when the polynomial crosses the x-axis inside the interval.

Example	Polynomial f(x)	Interval	Detected roots in interval	∫\|f(x)\|dx
1	x - 2	[0, 5]	2	6.5
2	x² - 4	[-3, 3]	-2, 2	15.333333
3	x² - 1	[-2, 2]	-1, 1	4
4	2x³ - x	[-1, 1]	-0.707107, 0, 0.707107	0.75

Formula used

This calculator evaluates the definite integral of an absolute-value polynomial over a chosen interval.

Core target: I = ∫_L^U |f(x)| dx

Polynomial model: f(x) = ax³ + bx² + cx + d

Antiderivative: F(x) = ax⁴/4 + bx³/3 + cx²/2 + dx

Piecewise absolute area: Split the interval at every detected root r. Then add |F(x_i) − F(x_i-1)| over each segment.

A Simpson-rule check is also computed numerically, so you can compare the exact piecewise result with a high-resolution numerical estimate.

How to use this calculator

Enter the polynomial coefficients a, b, c, and d.
Set the lower and upper integration bounds.
Choose decimal places for the displayed output.
Adjust graph points and root scan intervals if needed.
Press Calculate integral to generate the result.
Review the metric cards, piecewise table, and Plotly graph.
Use the CSV or PDF buttons to save the result.

FAQs

1) What does this calculator integrate?

It evaluates the definite integral of |f(x)| over a chosen interval. In this version, f(x) is a polynomial up to degree three entered through coefficients a, b, c, and d.

2) Why is the absolute value important?

Absolute value turns negative parts of the graph into positive area. That means the calculator measures total area between the curve and the x-axis, not cancellation between positive and negative regions.

3) How are roots used in the calculation?

The interval is split wherever f(x) crosses zero. Each subinterval keeps one sign, so the calculator can add the absolute area from each piece accurately.

4) What is the difference between signed and absolute integral?

The signed integral keeps negative values negative. The absolute integral converts every piece into positive contribution. When a curve crosses the axis, these two answers are usually different.

5) Why does the tool show a Simpson check?

The Simpson check is a numerical estimate of the same absolute-value integral. It helps confirm the piecewise result and is useful when you want an extra validation step.

6) Can I use decimals and negative bounds?

Yes. You can enter decimal coefficients, decimal bounds, and negative interval limits. The calculator handles those directly as long as the lower bound is less than the upper bound.

7) What does average absolute value mean?

It is the absolute integral divided by interval length. This gives the mean height of |f(x)| across the selected range and helps compare intervals of different widths.

8) Does this tool find an indefinite integral?

No. This page is built for definite integration over a chosen interval. It focuses on total absolute area, root-based splitting, graphing, and exportable worked output.