Question

Use geometry (i.e., areas of triangles, rectangles, and circles) to find the exact values of each of the definite integrals in Exercises $21-28$.

${\int }_{-r}^r\sqrt{r^2-x^2}dx$.

Short answer

The exact value of${\int }_{-r}^r\sqrt{r^2-x^2}dx$is,$\frac{{\mathrm{\pi r}}^2}{2}$.

Step-by-step solution

Step 1. Given information

${\int }_{-r}^r\sqrt{r^2-x^2}dx$.

Step 2. The following figure shows the graph of r2-x2.

Step 3. Find the area.

The shaded region can be considered to be a half circle of radius $r$units.

The area of the shaded half circle is,

$$\begin{gathered} A=\frac{\mathrm{\pi }{\left(\mathrm{radius}\right)}^2}{2} \\ =\frac{{\mathrm{\pi r}}^2}{2} \end{gathered}$$

Hence, the exact value is,$\frac{{\mathrm{\pi r}}^2}{2}$.