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Math Help - double integral Question

  1. #1
    Junior Member
    Joined
    Oct 2009
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    double integral Question

    Hi everyone,

    I am stuck on a few double integral questions.

    1. Let D be the region in the xy plane bounded by the curves r = cos(theta) and r = sin(2theta)/sqrt(2). Compute the double integral :

    Attachment 22892double integral Question-capturemath.png

    In Polar coordinates this region is described by two sets:

    A: theta from 0 to pi/4 r from 0 to (sin (2 theta)/ sqrt (2)
    B: theta from pi/4 to pi/2 from 0 to cos theta

    I tried the changed of variables formula, substituting x for r*cos (theta) and multiplying by the Jacobian r, but this does not give me a nice integration at all. Is there something I am doing wrong?


    2. Let D the region in the xy plane enclosed by x+1 = 1, x+y = 3, x = 0 and y = 0. Evaluate the integral:

    double integral Question-capturemath2.png

    Finding an Inverse Mapping: I found that the Jacobian for the inverse mapping from (x,y) to (u,v) is F^-1 (u,v) = ((u+v)/2, (u-v)/2) = (x,y). This gives me 1/2 as the absolute value of the Jacobian. And a region bounded by the lines u=1,u=3, v=u, v=-u.

    I then evaluate this integral using the image under F and the following bounds:
    v between u and -u, and u between 1 and 3.

    double integral Question-mathcapture3.png
    This of course does not give me a good integration. I think I am confused as to when I have to multiply by the Jacobian.

    Any insight would be helpful, Thanks in advance.
    Attached Thumbnails Attached Thumbnails double integral Question-capturemath3.png   double integral Question-capturemath2.png  
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