Chapter 5.2, Problem 54E

(a)

To determine

To evaluate: The value of integral ${\displaystyle \underset{0}{\overset{5}{\int }}f\left(x\right)dx}$ by using the facts $f\left(x\right)\ge 0$ on $\left[0,2\right]$, $f\left(x\right)\le 0$ on $\left[2,5\right]$, ${\displaystyle \underset{0}{\overset{2}{\int }}f\left(x\right)dx}=6$ and ${\displaystyle \underset{2}{\overset{5}{\int }}f\left(x\right)dx}=-8$.

(b)

To determine

To evaluate: The value of integral ${\displaystyle \underset{0}{\overset{5}{\int }}\left|f\left(x\right)\right|dx}$ by using the facts $f\left(x\right)\ge 0$ on $\left[0,2\right]$, $f\left(x\right)\le 0$ on $\left[2,5\right]$, ${\displaystyle \underset{0}{\overset{2}{\int }}f\left(x\right)dx}=6$ and ${\displaystyle \underset{2}{\overset{5}{\int }}f\left(x\right)dx}=-8$.

(c)

To determine

To evaluate: The value of integral ${\displaystyle \underset{2}{\overset{5}{\int }}4\left|f\left(x\right)\right|dx}$ by using the facts $f\left(x\right)\ge 0$ on $\left[0,2\right]$, $f\left(x\right)\le 0$ on $\left[2,5\right]$, ${\displaystyle \underset{0}{\overset{2}{\int }}f\left(x\right)dx}=6$ and ${\displaystyle \underset{2}{\overset{5}{\int }}f\left(x\right)dx}=-8$.

(d)

To determine

To evaluate: The value of integral ${\displaystyle \underset{0}{\overset{5}{\int }}\left(f\left(x\right)+\left|f\left(x\right)\right|\right)dx}$ by using the facts $f\left(x\right)\ge 0$ on $\left[0,2\right]$, $f\left(x\right)\le 0$ on $\left[2,5\right]$, ${\displaystyle \underset{0}{\overset{2}{\int }}f\left(x\right)dx}=6$ and ${\displaystyle \underset{2}{\overset{5}{\int }}f\left(x\right)dx}=-8$.