The temperature at the point (x, y, z) in a substance with conductivity K = 4.5 is u(x, y, z) = 5y2 + 5z2. Find the rate of heat flow inward across the cylindrical surface y2 + z2 = 7, 0 ≤ x ≤ 2.

The rate of the heat flow = 1260 π

Step-by-step explanation:

From the information given :

k = 4.5

u(x,y,z) = 5y² + 5z²

Surface cylinder:

y² +z² = 5,     0 ≤ x ≤ 2

Now parameterizing the surface by :

x = u  , y =  , z =

0 ≤ x ≤ 2        ,        0 ≤ t ≤ 2π

Taking integral of both equations; we have:

= 315 × (2) × (2π)

= 1260 π

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