# Thread: Cylindrical Coordinates Triple Integral

1. ## Cylindrical Coordinates Triple Integral

Hey everyone,

I am given the integral:

Evaluate the integral of f(x,y,z) = z(x^2 +y^2 +z^2)^(-3/2) over the part of the ball x^2 +y^2 +z^2 less than or equal to 81 defined by z greater than or equal to 4.5.

This is the top of the sphere. I know that theta runs from zero to 2pi, but I am very confused about how to find the angle phi, which runs from the z axis to a point P, and the distance from the origin to the point P, which we call Ro.

I need to find what they run between to calculate my integral. Any help would be appreciated. Thanks.

2. z= 4.5 intersects the sphere $\displaystyle x^2+ y^2+ z^2= 81$ where $\displaystyle x^2+ y^2+ (4.5)^2= 81$ or $\displaystyle x^2+ y^2= 81- 20.25= 60.75$, a circle with center at (0, 0, 4.5), radius $\displaystyle \sqrt{60.75}= 7.79$. That is, the straight line from the origin to a point on that circle is the hypotenuse of a right triangle with "opposite side" 4.5 and "near side" 7.79. phi is the arctan of $\displaystyle \frac{4.5}{7.79}$.