Question

Multivariable Calculus:
Find the volume of an ellipsoid with semiaxes a, b, and c by double integration, using either rectangular or polar coordinates.

I have the equation for the general (rectangular) form but would rather use polar to avoid crazy trig substitution. Thanks in advance!!
@thomaster @iambatman @e.mccormick @campbell_st

Answer 1

Right now I have the double integral in rectangular form as:
\[\int\limits_{-a}^{a} \left(\int\limits_{-b\sqrt{1-x^2/a^2}}^{b\sqrt{1-x^2/a^2}} c\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}dy\right)dx\]
Parentheses are just to make the limits of integration clear

Answer 2

ok your ellipsoid is
(x/a)^2 + (y/b)^2 + (z/c)^2 = 1

and the volume should be
V = 4/3 Pi * a*b*c

Answer 3

Yes, but I need to derive that using the double integral.

Answer 4

I wish I could just write down the answer and get credit

Answer 5

but we need an expression in terms of x^2 + y^2

Answer 6

I don't see an easy way to do it using polar coordinates.
but rectangular should work , or you might be able to use spherical coordinates

Answer 7

i agree with your integral, you just need to multiply it by 2 since you are finding the volume of the top half of the ellipsoid

Answer 8

would you like me to integrate that?

Answer 9

Yes, that would be awesome. I think I figured out how to set it up in polar form, but am struggling to integrate \[\frac{1}{\frac{\cos^2 \theta}{a^2} + \frac{\sin^2 \theta}{b^2}}\]

Answer 10

the double rectangular integral works*

Answer 11

make sure to put 2 * in front

Answer 12

Can you help me get started on the double rectangular integral?

Answer 13

I used a computer algebra system to check

Answer 14

its pretty hairy to integrate that

Answer 15

I know. but i have to. Which is why I'm trying to do it in polar.

Answer 16

right, the problem with the polar is that a^2 and b^2

Answer 17

I feel like if I can integrate that last part it will work. I'm gonna try if wolfram can do a step-by-step

Answer 18

http://www.wolframalpha.com/input/?i=simplify+2*%28int%28int%28c*sqrt%281-x^2%2Fa^2-y^2%2Fb^2%29%2C+y%3D+-b*sqrt%281-x^2%2Fa^2%29+..+b*sqrt%281-x^2%2Fa^2%29%29%2C+x+%3D+-a+..+a%29%29

Answer 19

not working...

Answer 20

Thanks. I have to go now, but will check back in to see what you came up with

Answer 21

I can solve this using triple coordinates and Jacobian transformation.

First use the transformation
x= a*u , y = b*v , z = c*w
then
J = Determinant{ {dx/du, 0, 0 }, {0, dy/dv, 0}, {0, 0, dz/dw} }
= DET { {a, 0, 0 }, {0, b, 0}, {0, 0, c} }
= abc

Therefore
V = int(int(int dx) dy)dz
over the ellipsoid (x^2/a^2 + y^2/b^2 + z^2/c^2 <= 1)
= (abc) int(int(int du)dv)dw
over the sphere (u^2 + v^2 + w^2 <= 1)

But volume of a sphere = 4/3 Pi r^3
= 4Pi/3 r^3 (abc)
here r = 1
= (4Pi / 3) abc

Answer 22

that looks correct to me