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The definite integral

The definite integral is the difference between the antiderivative at the upper and the lower limit. It gives the area between the graph and the $x$ -axis with sign: area below the axis counts as negative. You find an antiderivative, insert the upper limit and subtract the value at the lower one. The constant cancels, so it can be left out. For sine, an antiderivative is minus cosine.

Try it yourself

The integral as area

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

definite integral

\int \sin x\,dx = -\cos x + C

sine

Symbols

$a$	lower limit
$b$	upper limit
$F$	an antiderivative

Example

$\int_0^3 2x\,dx = [x^2]_0^3 = 9 - 0 = 9$ . $\int_0^{\pi} \sin x\,dx = [-\cos x]_0^{\pi} = 1 + 1 = 2$ .

Insert the upper limit first. An area below the axis becomes negative in the integral.

Practise integration for free →

← Antiderivative · Integration by parts →

Part of Mathematics R2: Integration.