How to Calculate the Volume of a Solid Using Polar Coordinates?

[Math10]

Homework Statement

Use polar coordinates to find the volume of the solid where T is the region that lies under the plane 3x+4y+z=12, above the xy-plane, and inside the cylinder x^2+y^2=2x.

Homework Equations

None.

The Attempt at a Solution

Here's my work:
x^2+y^2=2x
x^2-2x+y^2=0
x^2-2x+1+y^2=1
(x-1)^2+y^2=1
(r*cos(theta)-1)^2+(r*sin(theta))^2=1
r^2(cos(theta))^2-2r*cos(theta)+1+r^2(sin(theta))^2=1
r^2-2r*cos(theta)=0
factor
r(r-2*cos(theta))=0
r-2cos(theta)=0
r=2cos(theta)
V=r dz dr d(theta) from 0 to 2pi, from 0 to 2cos(theta), from 0 to 12-3r(cos(theta))-4r(sin(theta))
=18pi
But the answer in the book is 9pi. Which answer is right?

 

[LCKurtz]

Homework Statement

Use polar coordinates to find the volume of the solid where T is the region that lies under the plane 3x+4y+z=12, above the xy-plane, and inside the cylinder x^2+y^2=2x.

Homework Equations

None.

The Attempt at a Solution

Here's my work:
x^2+y^2=2x
x^2-2x+y^2=0
x^2-2x+1+y^2=1
(x-1)^2+y^2=1
(r*cos(theta)-1)^2+(r*sin(theta))^2=1
r^2(cos(theta))^2-2r*cos(theta)+1+r^2(sin(theta))^2=1
r^2-2r*cos(theta)=0
factor
r(r-2*cos(theta))=0
r-2cos(theta)=0
r=2cos(theta)
V=r dz dr d(theta) from 0 to 2pi, from 0 to 2cos(theta), from 0 to 12-3r(cos(theta))-4r(sin(theta))
=18pi
But the answer in the book is 9pi. Which answer is right?

The book is. Check your ##\theta## limits.

 

[Math10]

So how do I find the theta limits? Is it from -2pi to 2pi?

 

[LCKurtz]

So how do I find the theta limits? Is it from -2pi to 2pi?

Plot the graph of the polar equation circle and see what ##\theta## you need to get the circle.