Having problems setting integration limits due to symmetry.
I have two questions:
One asks to find the volume of the solid formed by the interior of the circle
r = cos(theta) capped by the plane z = x.
Because x = rcos(theta) and r = cos(theta), we have z = (cos(theta))^2
= (1+ cos2(theta))/2, which is the function we integrate.
The volume we wish to find is the right hand side of the lemniscate which is enclosed inside the circle.
For limits for the dr expression we can integrate from b = 1 to a = 0.
For the d(theta) limits I get confused. I was told to integrate the smallest interval possible and multiply by any symmetry factors. With that said, I can take beta = pi/2 and alpha = 0 and multiply by two due to symmetry about the x-axis. However, I know this is not the smallest interval as there must be a ray that caps the lemniscate function. I know how to find this ray, essentially a value for theta, by setting functios r equal to each other and finding that theta. However, I have an expression in terms of r and in terms of z.
My confusion in this next question is much the same as the above:
The question asks to find the volume of the solid based on the interior of the cardioid r = 1 + cos(theta), capped by the cone z = 2 - r.
We have z = 2 - (1 + cos(theta)) = 1 - cos(theta). Essentially another cardioid with the same size but with the cusp pointing in the opposite direction. The graph I get is basically an infinity sign along the y-axis, that is with rays pi/2 and 3pi/2, and with vertices (1,0) and (-1,0).
Again I get confused by symmetry due to the negative and positive contributions of the polar graph. For the d(theta) limits I believe I can integrate from alpha = 0 to beta = pi/2 and multiply by four due to symmetry. I get confused for the dr limits as the solid formed by the two cardioids expands over all four quadrants. Do I integrate from a = o to b = 2 and multiply by two due to symmetry? Two is the intersection point of the
1 + cos (theta) cardiod on the x-axis. I really get confused with this question as I imagine you can integrate from 0 to pi/2, pi/2 to pi, pi to 3pi/2 and 3pi/2 to 2pi breaking up the four quadrants and adding up the volumes of the four regions of the solid.
I would love any hints and any generalizations about symmetry considerations with polar coordinates.