double integral in polar coordinates, cant figure out how to find the limits of theta

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• April 19th 2012, 06:06 PM
AlmostFreed
double integral in polar coordinates, cant figure out how to find the limits of theta
Just trying to find what theta is between for the boundary (x^2)+(y^2)-y=0 in polar coordinates

the whole problem is:

Evaluate the surface area of the portion of the sphere (x^2)+(y^2)+(z^2)=1 that is inside the cylinder (x^2)+(y^2)-y=0. Hint; the double integral can be evaluated in polar coordinates

I put the sphere in polar coordinates and got the double integral of ((1-r^2)^-1/2) r d(r) d(theta) which is correct according to my answer sheet.

Then to find the limits of r I put the cylinder in polar coordinates and got r(r-sin(theta)) = 0. So 0<=r<=sin(theta)

But how would I find the limits of theta for the circle in the xy-plane (x^2)+(y-(1/2))^2 = 1/4 ? (I believe that circle equation is equivalent after completing the square)
• April 19th 2012, 06:58 PM
Prove It
Re: double integral in polar coordinates, cant figure out how to find the limits of t
Quote:

Originally Posted by AlmostFreed
Just trying to find what theta is between for the boundary (x^2)+(y^2)-y=0 in polar coordinates

the whole problem is:

Evaluate the surface area of the portion of the sphere (x^2)+(y^2)+(z^2)=1 that is inside the cylinder (x^2)+(y^2)-y=0. Hint; the double integral can be evaluated in polar coordinates

I put the sphere in polar coordinates and got the double integral of ((1-r^2)^-1/2) r d(r) d(theta) which is correct according to my answer sheet.

Then to find the limits of r I put the cylinder in polar coordinates and got r(r-sin(theta)) = 0. So 0<=r<=sin(theta)

But how would I find the limits of theta for the circle in the xy-plane (x^2)+(y-(1/2))^2 = 1/4 ? (I believe that circle equation is equivalent after completing the square)

\displaystyle \begin{align*} x^2 + y^2 - y &= 0 \\ x^2 + y^2 - y + \left(-\frac{1}{2}\right)^2 &= \left(-\frac{1}{2}\right)^2 \\ x^2 + \left(y - \frac{1}{2}\right)^2 &= \frac{1}{4} \end{align*}

Does this help you?
• April 19th 2012, 07:04 PM
AlmostFreed
Re: double integral in polar coordinates, cant figure out how to find the limits of t
Yes, but in a polar coordinate system, what would the values of theta be between? At what angle from the positive x-axis does a line extending from the origin first hit the curve, and at what angle is the line tangent to side of the curve in the 2nd quadrant?
• April 19th 2012, 07:05 PM
Prove It
Re: double integral in polar coordinates, cant figure out how to find the limits of t
Quote:

Originally Posted by AlmostFreed
Yes, but in a polar coordinate system, what would the values of theta be between? At what angle from the positive x-axis does a line extending from the origin first hit the curve, and at what angle is the line tangent to side of the curve in the 2nd quadrant?

It's a circle, what values of \displaystyle \begin{align*} \theta \end{align*} would a complete circle be swept through?
• April 19th 2012, 07:09 PM
AlmostFreed
Re: double integral in polar coordinates, cant figure out how to find the limits of t
If the circle contained the origin I know it would be 2(pi), but the circle is above the origin I can't seem to find where theta begins. I might be missing a simple concept here but I am stuck
• April 19th 2012, 07:15 PM
Prove It
Re: double integral in polar coordinates, cant figure out how to find the limits of t
Quote:

Originally Posted by AlmostFreed
If the circle contained the origin I know it would be 2(pi), but the circle is above the origin I can't seem to find where theta begins. I might be missing a simple concept here but I am stuck

You are missing a VERY simple concept. What angle does ANY circle make?
• April 19th 2012, 07:33 PM
AlmostFreed
Re: double integral in polar coordinates, cant figure out how to find the limits of t
Attachment 23664

Attachment 23665

I'm not talking about an angle from the center of the circle, I'm talking about "in a polar coordinate system, what would the values of theta be between? At what angle from the positive x-axis does a line extending from the origin first hit the curve, and at what angle is the line tangent to side of the curve in the 2nd quadrant?"