Evaluate $\iint _ { S } f ( x , y , z ) d S$
$f ( x , y , z ) = z ; S$ is the part of the torus with vector representation
$$\mathbf { r } ( u , v ) = ( 5 + 3 \cos v ) \cos u \mathbf { i } + ( 5 + 3 \cos v ) \sin \imath \mathbf { j } + 3 \sin v \mathbf { k } , 0 \leq u \leq 2 \pi , 0 \leq v \leq \frac { \pi } { 2 } .$$
Select the correct answer.
A) 0
B) $117 \pi$
C) 117
D) $27 \pi$

Question

Quizplus · Accepted Answer

The integral evaluates the surface integral of $z$ over the specified part of the torus. The symmetry and limits of integration result in the value $117 \pi$.

Evaluate ∬Sf(x,y,z)dS\iint _ { S } f ( x , y , z ) d S∬S​f(x,y,z)dS

Evaluate $\iint _ { S } f ( x , y , z ) d S$