Question

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

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

Short answer

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

Step-by-step solution

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-x2.

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,$18.283$.