Parametric Surfaces and Their Areas

[zm500]

Homework Statement

Find the area of the part of the sphere x^2 + y^2 + z^2 = a^2(a > 0constant) that lies inside the cylinder x^2 + y^2 = ax.

Homework Equations

double integral of the cross product of the vector Ra and Rb with respect to dA.

The Attempt at a Solution

I tried to parametrize it using spherical coordinates for the position vector R(a,b). I let a = 1, but i got confused trying to find the bound for phi and theta.

 

[LCKurtz]

Homework Statement

Find the area of the part of the sphere x^2 + y^2 + z^2 = a^2(a > 0constant) that lies inside the cylinder x^2 + y^2 = ax.

Homework Equations

double integral of the cross product of the vector Ra and Rb with respect to dA.

You mean double integral of the magnitude of the cross product.

The Attempt at a Solution

I tried to parametrize it using spherical coordinates for the position vector R(a,b). I let a = 1, but i got confused trying to find the bound for phi and theta.

If you complete the square on the cylinder and convert the resulting equation to cylindrical coordinates, see if you can show the cylinder becomes

r = a cos(θ)

Now, in spherical coordinates, r = ρ sin(φ) = a sin(φ). So if you have a point on the intersection of the sphere and cylinder, these r values must be equal giving:

sin(φ) = cos(θ).

To do the problem in spherical coordinates you need to use this relationship for the upper limit on the inner integral. Alternatively and which may be easier, you could use cylindrical coordinates for the parameterization of the sphere. That would certainly make the limits easier.

[Edit] After looking at it some more, it isn't easier in cylindrical although it works OK. You can use the last equation above to express φ in terms of θ for your inner limits, at least for first quadrant angles. Stick to the first quadrant and double to get your answer. It's easy if you understand the r and θ limits for your cylinder.