Solve the problem.
-$$\text { Find the average height of the paraboloid } z = 6 x ^ { 2 } + 9 y ^ { 2 } \text { above the disk } x ^ { 2 } + y ^ { 2 } \leq 16 \text { in the } x y - \text { plane. }$$
A) 60
B) 96
C) 84
D) 40

Question

Quizplus · Accepted Answer

The average height is found by integrating the function over the disk and dividing by the area of the disk. The integral of $ z = 6x^2 + 9y^2 $ over the disk $ x^2 + y^2 \leq 16 $ is $ \int_0^{2\pi} \int_0^4 (6r^2\cos^2\theta + 9r^2\sin^2\theta)r \, dr \, d\theta $. Solving this gives the total volume, and dividing by the area of the disk, $ 16\pi $, gives the average height as 60.

Solve the Problem A) 60 B) 96 C) 84 D) 40

Solve the Problem A) 60
B) 96
C) 84
D) 40