Question

Question #678c2

Answer 1

The ellipsoid has surface area `pi^2ab`.

Explanation:

The ellipsoid is generated by the circles drawn into the figure, which form loops around the x-axis. So, the surface area should be equal to the sum of the circumferences of each circle.

The radius of each circle is given by its height `y`. The radii of the circles range between `0` and `b`.

We should write a function describing this. Recalling the equation for ellipses, note that the function here is `x^2/a^2+y^2/b^2=1`.

This yields the simplification: `y^2=b^2(1-x^2/a^2)=b^2/a^2(a^2-x^2)`.

The height is then given by `y=b/asqrt(a^2-x^2)`.

This is the radius of the circle, so our circumference will be given at each point by `C=2pir=2piy=(2pib)/asqrt(a^2-x^2)`.

So, all we need to do is add up these circumferences. We can find the circles just to the right of the y-axis by integrating from `0` to `a` then doubling that for the surface area of the entire ellipsoid. So the surface area `S` is:

`S=2int_0^a(2pib)/asqrt(a^2-x^2)dx=(4pib)/aint_0^asqrt(a^2-x^2)dx`

To perform this integration, we could do some tedious work with the substitution `x=asintheta`.

A quicker method, however, would be to realize this is integral describing the area under the curve `y=sqrt(a^2-x^2)`, which is the same as the circle centered at the origin with radius `a`, `x^2+y^2=a^2`.

We care only for the positive part (the square root is positive) and from `0` to `a`, which is the first quadrant, or a fourth of the circle. A fourth of the area of the circle is `(pia^2)/4`.

Then:

`S=(4pib)/a((pia^2)/4)=pi^2ab`

Answer 2

The surface area of the torus is `4pi^2ab`.

Explanation:

The torus is formed by swinging a circle of radius `b` in a larger circle of radius `a`.

The surface area of the torus will be the collective sum of all the circumferences of the circles with radius `b`, where `C=2pib`.

Well, we're doing this along the entire distance of `2pia`, which is the circumference of the circle we're sweeping the littler circle along.

So the surface area would simply be `2pib(2pia)=4pi^2ab`.