Q. 28 - 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 $21-28$.

${\int }_{-2}^2\left(3+\sqrt{4-x^2}\right)dx$.

Step-by-Step Solution

Answer

The exact value of ${\int }_{-2}^2\left(3+\sqrt{4-x^2}\right)dx$ is, $18.283$.

Step 1 . Given information

${\int }_{-2}^2\left(3+\sqrt{4-x^2}\right)dx$.

Step 2 . The following figure shows the graph of 3 + 4 - x 2 .

Step 3 . Find the area.

The area of the shaded half circle and rectangle is,

$$\begin{gathered} \left(\frac{\pi (2{)}^2}{2}\right)+(3\times 4)=2\mathrm{\pi }+12 \\ =18.283 \end{gathered}$$

Therefore, the exact value is, $18.283$.