Question

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

${\int }_0^3(1-|x-1\left|\right)dx$.

Short answer

The exact value of${\int }_0^3(1-|x-1\left|\right)dx$is,$\frac{1}{2}$.

Step-by-step solution

Step 1. Given information

${\int }_0^3(1-|x-1\left|\right)dx$.

Step 2. The following figure shows the graph of (1-|x-1|).

Step 3. Find the area.

The area of the shaded triangle is,

$$\begin{gathered} \left(\frac{1}{2}\times 1\times 2\right)-\left(\frac{1}{2}\times 2\times \frac{1}{2}\right)=1-\frac{1}{2} \\ =\frac{1}{2} \end{gathered}$$

Therefore, the exact value is,$\frac{1}{2}$.