Integration and the area under a curve

Mathematics · Senior school calculus and first-year maths · about 10 min

Integration is differentiation in reverse, and it adds things up: the area under a curve, the distance travelled from a speed graph, the total work done by a changing force. The Fundamental Theorem of Calculus connects the two ideas.

The key idea

If $F'(x) = f(x)$ , then the definite integral from $a$ to $b$ is

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

and for powers, $\displaystyle\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$ ).

An x-y graph with a smooth rising curve above the x axis in dark blue. The area between the curve and the x axis, from x = a to x = b, is shaded pale blue. Thin vertical lines mark a and b. — A definite integral gives the area under the curve between two limits.

Try it yourself

Add more rectangles and watch the Riemann sum close in on the exact area.

Indefinite and definite integrals

An indefinite integral is a family of functions: $\int 2x\,dx = x^2 + C$ . The $+C$ is there because any constant disappears when you differentiate.
A definite integral has limits and gives a number: $\int_0^3 2x\,dx = 9 - 0 = 9$ . The constants cancel, so no $+C$ .

Area, and when it goes negative

For a curve above the $x$ -axis, $\int_a^b f(x)\,dx$ is the area between the curve and the axis. Parts below the axis count as negative. To find total area when a curve crosses the axis, split the integral at the roots and add the sizes of each part.

An x-y graph with a smooth wave in dark blue that starts above the x axis, crosses it once and ends below it. The area above the axis is shaded pale blue and the area below the axis is shaded pale red. — Area below the axis counts as negative.

Worked example: a definite integral

Evaluate $\displaystyle\int_0^2 (3x^2 + 2x)\,dx$ .

Show the solution

Antiderivative: $\int (3x^2 + 2x)\,dx = x^3 + x^2$

Apply the limits:

\left[x^3 + x^2\right]_0^2 = (8 + 4) - (0 + 0) = 12

Since $3x^2 + 2x \geq 0$ between $0$ and $2$ , this is the area under the curve: $12$ square units.

Common mistakes

Forgetting + C on an indefinite integral

Indefinite integrals always need $+C$ . Definite ones don't, because it cancels.

Integrating $x^n$ as $n x^{n-1}$

That's differentiating. Integrating raises the power and divides by the new power: $\frac{x^{n+1}}{n+1}$ .

Treating area below the axis as positive

The integral counts it as negative. For total area, split at the roots and add the absolute values.

Check your understanding

1. What is $\displaystyle\int x^3\,dx$ ?

2. What is $\displaystyle\int_1^3 2x\,dx$ ?

3. A speed-time graph is integrated over a time interval. What does the result give?

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