Q. 26 - Calculus

Question

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

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

Step-by-Step Solution

Answer

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

Step 1 . Given information

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

Step 2 . The following figure shows the graph of r 2 - x 2 .

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, $\frac{{\mathrm{\pi r}}^2}{2}$.