The temperature at the point $( x , y , z )$ in a substance with conductivity $K$ is $u ( x , y , z ) = 6 x ^ { 2 } + 6 y ^ { 2 }$ Find the rate of heat flow inward across the cylindrical $y ^ { 2 } + z ^ { 2 } = 6 , \quad 0 \leq x \leq 5$
A) $2530 \pi$
B) $2560 \pi$
C) $2520$
D) $2540 \pi$
E) $2520 \pi$

Question

Quizplus · Accepted Answer

The rate of heat flow is calculated using the formula $-K \int \nabla u \cdot \mathbf{n} \, dS$. The gradient $\nabla u = (12x, 12y, 0)$ and the normal vector $\mathbf{n}$ to the cylindrical surface is $(0, y, z)$. Evaluating the integral over the surface gives $2520 \pi$.

The Temperature at the Point (x,y,z)( x , y , z )(x,y,z)

The Temperature at the Point $( x , y , z )$