Add up a changing quantity.

A definite integral is the limit of sums of function heights times small widths. If F′ = f, the fundamental theorem evaluates the integral from a to b as F(b) − F(a). The constant cancels.

The integral is signed. Portions below the x-axis contribute negatively. Reversing the limits reverses the sign; equal limits give zero. A positive function gives ordinary area, while a function that changes sign may require splitting the interval to find total area.

INTERACTIVE MODEL

Try it. Watch it change.

f(x)

x^{2}

FunctionUpper limit1Trapezoids8

Integral = 0.3333 · approximation = 0.3359 · total area = 0.3333

Trapezoids approximate the signed integral. Total area adds the magnitudes of regions above and below the axis.

Explore: Increase the number of strips. Compare the approximation with the exact integral.

Fundamental theorem of calculus

\int_a^b f(x)\,\mathrm dx=F(b)-F(a)

WORKED EXAMPLE

Evaluate ∫₀² $3x^{2}$ dx.

An antiderivative is $x^{3}$ .
F(2) − F(0) = 8 − 0 = 8.

Assumed knowledge

Antiderivatives and limits.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Use the fundamental theorem of calculus.
Connect Riemann sums to signed area.

QCAA Mathematical Methods 2025 v1.3 · p. 28 ↗

READY TO TRY IT?

Add up a changing quantity.

Evaluate ∫₀² 3x23x^{2}3x2 dx.

Put the idea to work.

Evaluate ∫₀² $3x^{2}$ dx.