Double Integral - Going from Cartesian to Polar

[erok81]

Homework Statement

See attachment.

Change the Cartesian integral into an equivalent polar integral, then evaluate the integral.

I have no problems at all converting the actual function I am integrating or the integration itself, it is just the limits I cannot do.

I've posted two examples in my attachment. The first one I have the answer for and I can get three of them but the 6csc theta I cannot figure how to get that. The other I've posted the three I can do (but don't know if it is right because I have no answer for this one) but still that fourth one I can't get get.

I think the problem is our teacher gave us an assignment to convert Cartesian to cylindrical to spherical but didn't explain how to do it or give us any answers. You can imagine that was a total waste of time - "here do this assignment but I am not going to tell you how or give you the answers."

 

[Dick]

Why don't you sketch a graph of the region in the x-y plane? Then the theta limits of pi/4 to pi/2 shouldn't be any surprise. Now you want to find the r limits as a function of theta. y=r*sin(theta). The limits for y are 0 and 6. Solve for r. An upper limit of 6*csc(theta) shouldn't be a surprise either.

 

[erok81]

That makes sense.

I guess the part I am confused about is the cartesian coordinates are integrated as dx dy so I though you'd covert dx to dr and dy to d(theta) since they are in that order.

I thought I got it but after I read it I took my dog out and came back to work on it and now it doesn't make sense.

 

[erok81]

Or is it just in this case you know that y=rsin(theta) so you can pull the y values from the other integral limits?

 

[erok81]

Oh...maybe I do get it.

In my second example is the upper dr limit 2sec(theta)?

Now that I think about it more, I remember the teacher said carve with r and sweep with theta. If I do that my post #4 is irrelevant and that part makes sense.

 

[Dick]

Oh...maybe I do get it.

In my second example is the upper dr limit 2sec(theta)?

Now that I think about it more, I remember the teacher said carve with r and sweep with theta. If I do that my post #4 is irrelevant and that part makes sense.

Yes, to 2*sec(theta). But in the second example I don't think the theta limits are right. Did you draw the region?