- #1

mishima

- 570

- 36

## Homework Statement

This is a problem from Boas, Mathematical Methods of the Physical Sciences chapter 5, section 5, number 6.

Find the area of the cylinder x^2+y^2-y=0 inside the sphere x^2+y^2+z^2=1.

## Homework Equations

This section deals with projecting curved areas onto a coordinate plane and evaluating using double integrals. The projected surface is related to the curved surface by

dxdy=dA cos γ

or

dA=sec γ dxdy

where γ is measured off the 3rd axis (the one not being projected onto). Also,

sec γ = grad φ / (dφ/dz)

where φ is the equation for the curved surface. So, the differential area can be evaluated through a double integral after choosing the right bounds.

## The Attempt at a Solution

In this problem, the cylinder axis is parallel to the z axis, so a projection onto the xy plane will not work. Instead, we can cycle through the axes and modify the equations like:

dA=sec γ dydz

sec γ = grad φ / (dφ/dx)

Where the latter becomes, for our cylinder,

sec γ = 1/(y-y^2)^(1/2)

The equation for the sphere gives the bounds for the double integral, y going from -(1-z^2)^(1/2) and +(1-z^2)^(1/2) while z is just from 0 to 1. I can evaluate one, but not both of the integrals with what I know. I have also tried changing coordinate systems to polar but there was not a simplification.

The answer is supposedly 4, so I feel like I'm missing something obvious.