Question

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

${\int }_{-2}^4|3x+1|dx$.

Short answer

The exact value of${\int }_{-2}^4|3x+1|dx$is,$28.97$.

Step-by-step solution

Step 1. Given information

${\int }_{-2}^4|3x+1|dx$.

Step 2. The following figure shows the graph of 3x+1.

Step 3. Find the area.

The area of the shaded triangles is,

$$\begin{gathered} \left(\frac{1}{2}\times 0.33\times 5\right)+\left(\frac{1}{2}\times 13\times 4.33\right)=0.825+28.145 \\ =28.97 \end{gathered}$$

Therefore, the exact value is,$28.97$.