Question

Average height of a hemisphere Find the average height of the hemispherical surface z = √(a²-x²-y²) above disk x²+y²< a² in the xy-plane.

Answer 1

To find the average height of the hemispherical surface z = √(a²-x²-y²) above the disk x²+y² < a² in the xy-plane, we need to find the average value of z over the disk. This can be done by finding the double integral of z over the disk, divided by the area of the disk.

Step-by-step explanation:

To find the average height of the hemispherical surface z = √(a²-x²-y²) above the disk x²+y² < a² in the xy-plane, we need to find the average value of z over the disk. We can do this by finding the double integral of z over the disk, divided by the area of the disk.

Let's denote the region of the disk as D. The average height H is given by:

H = (1/Area(D)) ∬D z dA = (1/Area(D)) ∬D √(a²-x²-y²) dA

By using polar coordinates, we can rewrite the double integral as:

H = (1/Area(D)) ∬D √(a²-r²) r dr dθ

Now you can calculate the average height of the hemispherical surface by evaluating this integral over the disk using appropriate limits.