The temperature at the point $( x , y , z )$ in a substance with conductivity $K = 4$ is $u ( x , y , z ) = 2 x ^ { 2 } + 2 y ^ { 2 }$.

Find the rate of heat flow inward across the cylindrical $y ^ { 2 } + z ^ { 2 } = 2,0 \leq x \leq 7$.
Select the correct answer.
A) 224
B) $234 \pi$
C) $244 \pi$
D) $264 \pi$
E) $224 \pi$

Question

The temperature at the point $( x , y , z )$ in a substance with conductivity $K = 4$ is $u ( x , y , z ) = 2 x ^ { 2 } + 2 y ^ { 2 }$.

Find the rate of heat flow inward across the cylindrical $y ^ { 2 } + z ^ { 2 } = 2,0 \leq x \leq 7$.
Select the correct answer.
A) 224
B) $234 \pi$
C) $244 \pi$
D) $264 \pi$
E) $224 \pi$

Quizplus · Accepted Answer

The rate of heat flow is calculated using the formula $-K \int \nabla u \cdot dS$. The gradient $\nabla u = (4x, 4y, 0)$. The surface normal for the cylindrical surface is $(0, y, z)$. The integral evaluates to $224 \pi$.

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

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