1. ## Integration using transformations

Any ideas how I go about starting this problem. I cant find anything relevant in my notes..

Use the transformation
x = au, y = bv, z = cw to evaluate the integral

integral integral integral(R) 1
dV

where
R is the region bounded by the ellipsoid

(x^2/a^2) + (y^2/b^2) + (z^2/c^2) = 1

and hence calculate the volume of R.

Any help would be much appreciated

2. Originally Posted by SirOJ
Any ideas how I go about starting this problem. I cant find anything relevant in my notes..

Use the transformation x = au, y = bv, z = cw to evaluate the integral

integral integral integral(R) 1
dV

where

R is the region bounded by the ellipsoid

(x^2/a^2) + (y^2/b^2) + (z^2/c^2) = 1

and hence calculate the volume of R.

Any help would be much appreciated
If V is the volume of the ellipsoid and U the volume of a sphere of radius 1 (which is what you get after your transformation) then

$\displaystyle \iiint_{V} 1 dV = \iiint_{U}1 \frac{\partial(x,y,z)}{\partial(u,v,w)}\, dU$

where $\displaystyle \frac{\partial(x,y,z)}{\partial(u,v,w)}$ is the Jacobian of the transformation and equals $\displaystyle abc$. Thus,

$\displaystyle \iiint_{V} 1 dV = a b c \iiint_{U}1 \, dU = \frac{4}{3} \pi a b c$.

3. Thanks alot for the reply... I'm not sure i really understand the answer to my last question. Is there any way you could step through the following question..?

Use the variable transformation
u = x y and v = x + y to evaluate the integral integral(S) sin(x y) cos(x + y) dA,

where S is the square with vertices (0, 0), (pi/2, pi/2), (pi/2,-pi/2) and (pi, 0).

Thank,
SirOJ