Homework Help: Double integral change of variable polar coordinates question

1. Oct 1, 2014

jamesdocherty

1. The problem statement, all variables and given/known data

evaluate the double integral of cosh(6x^2+10y^2) dxdy by making the change of variables x=rcos(theta)/sqrt(3) and y=rsin(theta)/sqrt(5)

let D be the region enclosed by the ellipse 3x^2+5y^2=1 and the line x=0 where x>0.

2. Relevant equations

3. The attempt at a solution

first I found the Jacobian, using the determinant of [dx/dr dx/dtheta, dy/dr dy/dtheta] that turned into [cos(theta)/sqrt(3) -rsin(theta)/sqrt(3), sin(theta)/sqrt(5) rcos(theta)/sqrt(5)] to be r/sqrt(15)

then after that i found out what the region in the polar coordinates would be covered, i subbed in x=rcos(theta)/sqrt(3) and y=rsin(theta)/sqrt(5) to find out r is either 1 or -1, as it must be positive it has to be 1. And as it's only half the ellispe on the postive x-axis, theta goes from -pi/2 to pi/2. lastly i subbed in x=rcos(theta)/sqrt(3) and y=rsin(theta)/sqrt(5) into the equation to obtain cosh(2r^2).

Hence now the integral is 1/sqrt(15) * double integral rcosh(2r^2) drdtheta, where 0<r<1 and -pi/2<theta<pi/2 and the 1/sqrt(15) and the r into of cosh is in there due to the jacobian.

integrating this integral i obtained the answer to be pi*sinh(2)/4sqrt(15) which isnt the right answer, i integrated rcosh(2r^2) to be equal to sinh(2r^2)/4 which says is correct on wolfman alpha and then i had 1/sqrt(15)* integral sinh(2)/4 dtheta which turned into 1/sqrt(15)*[theta*sinh(2)/4] for pi/2 and -pi/2 which come out to be pi*sinh(2)/4sqrt(15), i am very confused as why this isnt the right answer and any help would be much appreciated.

MY ANSWER I GOT: pi*sinh(2)/4sqrt(15) (WHICH IS WRONG)

Last edited by a moderator: Oct 1, 2014
2. Oct 1, 2014

vela

Staff Emeritus
Looks okay to me. Why do you think your answer is wrong?

3. Oct 1, 2014

Staff: Mentor

Minor point: "intergal" and "intergate" are not words.

4. Oct 2, 2014

jamesdocherty

the answer in the question booklet is pi*sinh(2)/4 not pi*sinh(2)/4sqrt(15), do u think there could be an error in the question booklet?

5. Oct 2, 2014

vela

Staff Emeritus
Yup.