Consider the surfaces given by $$ M:(x^2+z^2-z)^2=(x^2+z^2) $$ $$ N:y^2=x^2+z^2 $$

(a) Notice that since there is no y present in M the shape of the surface is a cylinder determined by its shape in the xz-plane! Convert M to polar coordinates and sketch the surface. What’s the name of the type of curve that determines its shape? Note: You may assume r not euqal 0.

(b) Sketch the solid bounded by M, N and the xz-plane.

(c) Find the volume of the solid by setting up an appropriate triple integral and computing. Hint: Polar! Also, some functions are odd.

a) I got $$r=0$$ and $$r=1+sin \theta $$

and i know the shape of r=1+sin theta is a cardioid

b) I know N is a elliptical cone, but I dont know how to sketch them on a graph, and I have no idea how the solid would look like.

I am stuck afterward...


$x = r\cos\theta\\ z = r\sin \theta\\ y= y$


$dx\ dy\ dz = r\ dy\ dr\ d\theta$

Regarding $M,$ as you say $r = 1 + \sin \theta$ which is a cartiod cylinder. $N$ is a double cone (not elliptical)

$y^2 = r^2\\ y = \pm r$

$V = \int_0^{2\pi}\int_0^{1-\sin\theta}\int_{-r}^{r} r \ dy\ dr\ d\theta$


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