Q. 27 - 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 $21-28$.

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

Step-by-Step Solution

Answer

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

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, $\frac{1}{2}$.