Evaluate the triple integral $\iiint _ { E } 1 d V$ in cylindrical coordinates, where $E = \{ ( r , \theta , z ) \mid 0 \leq r \leq 1,0 \leq \theta \leq \pi , 0 \leq z \leq 1 \}$ .
A) $\frac { \pi } { 4 }$

B) $\frac { \pi } { 3 }$

C) $\frac { \pi } { 2 }$

D) $\frac { 2 \pi } { 3 }$

E) $\frac { 3 \pi } { 4 }$

F)$\pi$

G) $\frac { 4 \pi } { 3 }$

H)2$\pi$

Question

Evaluate the triple integral $\iiint _ { E } 1 d V$ in cylindrical coordinates, where $E = \{ ( r , \theta , z ) \mid 0 \leq r \leq 1,0 \leq \theta \leq \pi , 0 \leq z \leq 1 \}$ .
A) $\frac { \pi } { 4 }$

B) $\frac { \pi } { 3 }$

C) $\frac { \pi } { 2 }$

D) $\frac { 2 \pi } { 3 }$

E) $\frac { 3 \pi } { 4 }$

F)$\pi$

G) $\frac { 4 \pi } { 3 }$

H)2$\pi$

Quizplus · Accepted Answer

The integral $\iiint _ { E } 1 d V$ in cylindrical coordinates can be evaluated as the product of the ranges of $r$, $\theta$, and $z$, which are $1$, $\pi$, and $1$, respectively. Therefore, the volume is $1 \times \pi \times 1 = \pi$, which corresponds to $\frac { \pi } { 2 }$.

Evaluate the Triple Integral ∭E1dV\iiint _ { E } 1 d V∭E​1dV

Evaluate the Triple Integral $\iiint _ { E } 1 d V$