Evaluate the integral.
-$$\int _ { 7 } ^ { 8 } \int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } \varrho \sin \varphi \mathrm { d } \theta \mathrm { d } \varphi \mathrm { de }$$
A) $\frac { 5 } { 2 }$
B) $\frac { 15 } { 4 }$
C) $\frac { 5 } { 2 } \pi$
D) $\frac { 15 } { 4 } \pi$

Question

Quizplus · Accepted Answer

The integral can be evaluated by first integrating with respect to $\theta$, which gives $\theta$ evaluated from $0$ to $\frac{\pi}{2}$, resulting in $\frac{\pi}{2}$. Then, integrating $\varrho \sin \varphi$ with respect to $\varphi$ from $0$ to $\frac{\pi}{2}$ gives $-\cos \varphi$ evaluated from $0$ to $\frac{\pi}{2}$, resulting in $1$. Finally, integrating $\varrho$ with respect to $\varrho$ from $7$ to $8$ gives $\frac{1}{2}\varrho^2$ evaluated from $7$ to $8$, resulting in $\frac{1}{2}(64-49) = \frac{15}{2}$. Multiplying all parts together gives $\frac{\pi}{2} \times 1 \times \frac{15}{2} = \frac{15}{4}\pi$.

Evaluate the Integral A) 52\frac { 5 } { 2 }25​

Evaluate the Integral A) $\frac { 5 } { 2 }$