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

Question

Quizplus · Accepted Answer

The integral evaluates the surface integral of the function $z$ over the specified part of the torus. The symmetry and bounds of the torus section lead to the result $117 \pi$.

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

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