Algebra

Integration

Explore signed area and families of antiderivatives.

Predict, test and explain

Step 1 of 3

Predict

What is the signed integral of f(x) = x from −1 to 1?

Further exploration: introductory polynomial calculus.

\int_{-2}^{2}(x^{2}-1)\,dx\approx 1.333

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F(x)=\frac{1}{3}x^{3}-x+C

\text{midpoint estimate}\approx 1.25

Green shading is above the axis; pink is below. The definite integral counts signed area, so contributions can cancel. Rectangles give a midpoint estimate. Evaluate F(upper) − F(lower); the constant C cancels.

Integral and estimate
Quantity	Value
Lower limit	-2
Upper limit	2
Definite integral	$\approx 1.333$
Midpoint estimate	1.25

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Predict

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Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

Evaluate this definite integral.

$\int_0^2 3x^2\,dx$

Hint

Find an antiderivative, then subtract its value at the lower limit from its value at the upper limit.

Reveal answer

An antiderivative is the input cubed; the upper value minus the lower value is eight.

$\int_0^2 3x^2\,dx=\left[x^3\right]_0^2=8$

Build an explanation

Hint 1

An indefinite integral gives a family of antiderivatives. Differentiating any member returns the original function, so include + C.

Hint 2

For a polynomial term axⁿ with a non-negative integer n, add 1 to the power and divide by the new power: axⁿ⁺¹/(n + 1).

Hint 3

For a definite integral from a to b, find an antiderivative F and calculate F(b) − F(a). This is signed area: regions below the x-axis contribute negatively.

Worked example

This example uses fixed values, separate from the diagram controls.

Find the indefinite integral of x² − 1, then its definite integral from x = 0 to x = 2.

An antiderivative of x² is x³/3; an antiderivative of −1 is −x.
The indefinite integral is x³/3 − x + C. Its derivative is x² − 1.
For the definite integral, use F(x) = x³/3 − x and evaluate F(2) − F(0).
(8/3 − 2) − 0 = 2/3. The constants cancel, so this definite integral has one numerical value.

Answer: Indefinite integral x³/3 − x + C; definite integral from 0 to 2 is 2/3.

Connect this idea

Watch out: A definite integral can be negative or zero even when regions have area. For total geometric area, split at x-axis crossings and add the positive magnitudes of each region.