Question

Part 1A: Why is ρ=8cos(φ) a sphere? (greek symbols are rho and phi). What is its center and radius? Algebra and trig will be needed.  

Part 1B: Given a right circular cone of radius R and height H, set up a triple integral in spherical coordinates to determine its volume. (Answer should be 1/3πr^2h)

Please clearly outline steps used to solve this problem. I have been stuck for a long time. Thank you in advance!

Answer 1

Problem (1A)

Recall the spherical coordinates

Then

Now, keep one as it is and replace the other by since .

This is the equation of a sphere centered at with radius units.

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Problem (1B)

To calculate the volume, we need to find the limits of integration.

Now, is one of the angles of a right angled triangle with sides and

So, on the edge of the cone.

Therefore,

Next,

Now, depends on .

We have,

since

Therefore, the volume of the cone is given by

where is the solid cone

Now, do a change of variable. Substitute so that the integral becomes