Calculus question?

Question

If anyone could explain how the following is done, it would be greatly appreciated!

Write the equation for the surface generated by revolving the given around the indicated axis.

4x + 9y^2 = 36 around the y-axis

Answer 1

a^2=36/4=9

a=3

b^2=36/9=4
b=2

Instead of an ellipse in a plane you will get a shape in space.
Only draw a circle center O between -3 and 3
like a sphere.

Answer 2

Okay, this is your equation:

4x + 9y^2 = 36

we can write it as x=f(y)

x = 9 - (9/4)y^2

which is convenient since we are going to revolve it arround the y-axis.

Now, this is the idea to calculate the surface area. Spin the curve, and then slice it horizontaly (parallel to x-axis).
What will be the surface area of this slice?
If you look at it, it will look like a circle, and the arclenght of the circle is 2* pi * radius
2 and pi are constants, and in our problem the radio is determined by f(y).
So we are going to calculate the revolution area of every infinitesimal slice and when we add it up, we will get the complete revolution area (it sounds a lot harder than what it is).
We are going to integrate (because integrals are really sums).

integral from a to b [(2 * pi * f(y))] dy

The curve has an a max in y=2, and i assume that the lower boundary will be the x-axis so a=0 b=2
Solve your integral and that's it.

Answer 3

Begin by isolating y.

9yy = 36 - 4x.

yy = (36 - 4x)/9.

y = ±√[(36 - 4x)/9].