Area under Curve/Examples/sin x from 0 to pi

Example of Use of Area under Curve

The area bounded by the curve $y = \sin x$ and the $x$-axis between $x = 0$ and $x = \pi$ is $2$.

Proof

Let $\AA$ be the area in question.

From Area under Curve we need to evaluate the definite integral:

$\AA = \ds \int_0^\pi \sin x \rd x$

So:

\(\ds \AA\)

\(=\)

\(\ds \int_0^\pi \sin x \rd x\)

\(\ds \)

\(=\)

\(\ds \bigintlimits {-\cos x} 0 \pi\)

Primitive of Sine Function

\(\ds \)

\(=\)

\(\ds \paren {-\cos \pi} - \paren {-\cos 0}\)

Definition of Definite Integral

\(\ds \)

\(=\)

\(\ds \paren {-\paren {-1} } - \paren {-1}\)

Cosine of Straight Angle, Cosine of $0 \degrees$

\(\ds \)

\(=\)

\(\ds 2\)

evaluation

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XV}$: $3$