1. Find the area of that portion of the cylinder lying inside the sphere in the first octant.
2. Find the area of that portion of the surface above the first quadrant of the -plane for which and .
I seriously have no idea how to do any of these.
If your surface is given parametrically as
then the surface area of S is given by
I think in the beginning it's easier to use cartesian coordinates and then switch to polar (if necessary) then trying to use this formula in general. If so, then
Hence the formula I gave earlier. Are you familar with this
I'll set you up on the first one - you try the second.
Here so and and
Now your region of integration is a circle so we switch to polar coords., i.e, (the circle) and
For the limits of integration, and . So the surface area is given by
which you should be able to evaluate.
1. You can rewrite the problem is terms of a dependent variable. Notice that while z is not a function of a given x,y pair, x is a function of a given z,y pair and y is a function of a given x,z pair. (If you restrict the range. Think of a circle in two dimensions y=+/-(1-x^2)^.5 ) This approach is somewhat difficult because you must discern this function.
2. Use vectors. This is much easier. Express the cylinder in terms of a vector valued function in two parameters. This way, the surface is a function relatively easily derived.
This was challenging. The parametric equation for the cylinder was complicated by its intersection with the sphere, but I managed it. Ok Check it.
On the z = 0 plane, x and y trace out a circle with radius two centered at x=0 y=2 Parametrically, that can be expressed as (2*cos u) i + (2 + 2*sin u) j Now, z is given by +/- (16-x^2-y^2)^.5 We'll work exclusively with the positive half and just double our answer because everything is symmetric with regard to z=0. Further, the range of z is constrained within the restricted domain of x and y. That is, x^2+(y-2)^2 must equal 4. That is after all, the cylinder that we are interested in. So, we are looking for the Z values that lie on the surface of the sphere and lie directly above that circle drawn out on the z=0 plane. We already have an equation for y given by the cylinder. That is, y= +/- (4-x^2)^.5 + 2 By substituting this equation into the equation for the top hemisphere of the sphere we get z in terms of x alone. Z=(16+/-4*(4-x^2)^.5 -8)^.5 Now, we already expressed x in terms of a parameter u above. So, the equation becomes Z= (16+/-4*(4-4*(cosu)^2)^.5 -8)^.5 Now, we have a parametric equation for a curve in space, but not a surface. For that, we introduce a second parameter v and apply it to the K vector. Further, we allow v to run from 0 to 1 and no further. So finally we have the parametric equation:
R(u,v) = (2 cosu)i + (2+2sinu)j + v* ((16+/-4*(4-4*(cosu)^2)^.5 -8)^.5)k
To determine the surface area, use the cross product of the two partial derivatives.
Now that I look at the expression for K again, I realize that it can be simplified significantly. It simplifies to v* 2*(2 +/- 2*sinu)^.5 After you determine the magnitude of the cross product of the two partial derivatives, the final integral that needs to be solved is just 4(2 +/- 2 sinu)^.5 dudv 2pi-0 and 1-0 respectively. I don't know how the +/- works out. I've never attempted it before.
Now that I think about it, are you sure that you have the problem correct? This seems a bit challenging for someone just learning surface area. Do you think that you mistransposed the problem? Integrating the sphere over the cylinder is MUCH easier and seems more like a beginer's problem than the cylinder within the sphere.
OK. I see what you've done. You have given the parametric for the cylinder, however, you've neglected the boundary. Your parametric cylinder is a complete cylinder that runs from z=4 to z= -4 You must realize that he originally asked for the surface area of the portion of this cylinder lying only within the given sphere. Think of what the intersection looks like. It's complicated. We're not talking about a plane here. The intersection with a sphere is going to be complex.