Evaluate the integral.
-
A) $3 \sqrt { 2 } \mathbf { i } + 9 \sqrt { 2 } \mathbf { j }$
B) $3 \sqrt { 2 } \mathrm { i }$
C) 0
D) $9 \sqrt { 2 } \mathrm { i }$

Question

Evaluate the integral.
-
A) $3 \sqrt { 2 } \mathbf { i } + 9 \sqrt { 2 } \mathbf { j }$ 
B) $3 \sqrt { 2 } \mathrm { i }$ 
C) 0 
D) $9 \sqrt { 2 } \mathrm { i }$

Quizplus · Accepted Answer

Since the cylinder is symmetric around the $z$-axis, we can use cylindrical coordinates. Let $x= r\cos{	heta}$, $y=r\sin{	heta}$, $z=z$, where $0 \leq 	heta \leq 2\pi$, $0 \leq r \leq 2\sqrt{2}$, and $0 \leq z \leq 3$. The integral becomes
$$\int_0^{2\pi} \int_0^{2\sqrt{2}} \int_0^3 r^2\sin{	heta} \cdot rdrd	heta dz.$$
Evaluating the inner integral, we have 
$$\int_0^3 r^2\sin{	heta} \cdot rdr = \frac{r^4}{4}\sin{	heta}\large|_0^{2\sqrt{2}} = 8\sqrt{2}\sin{	heta}.$$ 
Now, the integral becomes 
$$\int_0^{2\pi} \int_0^{2\sqrt{2}} 8\sqrt{2}\sin{	heta} \cdot rdrd	heta = 0,$$
since $\int_0^{2\pi} \sin{	heta} d	heta = 0$. Therefore, the answer is $\boxed{	extbf{(C) }0}.$

Evaluate the Integral 32i+92j3 \sqrt { 2 } \mathbf { i } + 9 \sqrt { 2 } \mathbf { j }32​i+92​j

Evaluate the Integral $3 \sqrt { 2 } \mathbf { i } + 9 \sqrt { 2 } \mathbf { j }$