Solve the problem.
-$$\text { Find the average height of the paraboloid } z = x ^ { 2 } + y ^ { 2 } \text { above the disk } x ^ { 2 } + y ^ { 2 } \leq 100 \text { in the } x y - \text { plane. }$$
A) 150
B) 1003
C) 50
D) 250

Question

Quizplus · Accepted Answer

The average height of the paraboloid $z = x^2 + y^2$ above the disk $x^2 + y^2 \leq 100$ can be found by integrating the function over the area of the disk and then dividing by the area of the disk. The area of the disk is $\pi r^2$, where $r = 10$, so the area is $100\pi$. The integral of $z = x^2 + y^2$ over the disk gives the total volume under the paraboloid and above the disk, which can be calculated using polar coordinates. The integral becomes $\int_0^{2\pi} \int_0^{10} r^2 \cdot r dr d\theta$, where $r^2$ comes from converting $x^2 + y^2$ into polar coordinates and the extra $r$ comes from the Jacobian determinant when converting to polar coordinates. Solving this integral gives $\frac{1}{2} \cdot 10000\pi$. Dividing this volume by the area of the disk ($100\pi$) gives the average height, which is $50$.

Solve the Problem A) 150 B) 1003 C) 50 D) 250

Solve the Problem A) 150
B) 1003
C) 50
D) 250