Definite Integral Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

Evaluate

\int_0^{\pi} \sin x\,dx

and explain the geometric meaning.

1
Antiderivative of $\sin x$ is $-\cos x$ .
2
Apply FTC: $[-\cos x]_0^{\pi} = -\cos\pi - (-\cos 0) = -(-1) - (-1) = 1 + 1 = 2$ .

Answer

2

[0,\pi]

\sin x \geq 0

, so the definite integral equals the actual geometric area under one arch of the sine curve. The area of that arch is exactly 2 square units.

About Definite Integral

An integral evaluated between specific bounds $a$ and $b$ , yielding a single number: the signed area under the curve.

Evaluate [formula].

Evaluate [formula] and interpret the sign of the result.

Evaluate [formula].