Question

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area

\(y = \sin x,0 \le x \le \pi \)

Short answer

The area of the region is 2.

Step-by-step solution

Graph.

The rough estimate of the area of the region and the graph below.

The rough estimate of the area is2 units.

Calculate the area.

The exact area can be found by evaluating the integral between0and \(\pi \)

\(\begin{aligned}{c}\int_0^\pi {\sin } xdx &= ( - \cos x)_0^\pi \\ &= - (\cos (\pi ) - \cos (0))\\ &= 1 + 1\\ &= 2\end{aligned}\)

Hence the area of the region is 2.