Question

WILL AWARD MEDAL AND FAN
Vector calculus. question attached

Answer 1

@iambatman

Answer 2

parameterise the cylinder

\[\vec r = <4 \cos \theta, 4 \sin \theta, z>\]

tangent vectors:
\[d \vec r_z = <0,0,1>dz\]
\[d \vec r_{\theta} = <-4 \sin \theta,4 \cos \theta,0>d \theta\]
\[d \vec{S} = d \vec r_{z} \space{ } \times \space{ } d \vec r_{\theta} = \left| \left[\begin{matrix} \hat i & \hat j & \hat k\\ 0 & 0 & 1\\-4 \sin \theta& 4 \cos \theta & 0\end{matrix}\right] \right|\]
\[d \vec{S} = -4 <\cos \theta, \sin \theta, 0> dz d \theta\]
\[dS = |d \vec S| = 4 dz d \theta\]
integral is
\[4 \int\limits_{0}^{2 \pi} \int\limits_{0}^{4} 16 + z^2 dz d \theta = 8 \pi \left[ 16 z + \frac{z^3}{3} \right]_0^4 = 682.6666 \pi\]

that seems to be the answer you tried yet it marked it wrong

because, i think, you also have to integrate over the end caps

just using \[dS = r dr d \theta\] as the area element
for the bottom cap, z = 0:
\[ \int\limits_{0}^{2 \pi} \int\limits_{0}^{4} r^3\ dr d \theta = 128 \pi\]

for the top cap at z = 4:
\[ \int\limits \limits_{0}^{2 \pi} \int\limits \limits_{0}^{4} r^3 + 16 \ dr d \theta = 256 \pi\]

total = 1066.6666*pi