Question #678c2

2 Answers
Nov 15, 2017

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#

Nov 15, 2017

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#.