ma215-080-f09 outline 2009-11-02

Last time we looked at setting up triple integrals in rectangular and cylindrical coordinates.
Example: Consider a quarter wedge, in the first octant and bounded by z = 5 - 2y and x² + y² = 4. If the joint density function for a blemish in the volume is P(x,y) = sqrt(x² + y²)/125, then the probability of there being a blemish in the volume is the triple integral of P over the volume.
Game: Set up this integral in rectangular and cylindrical coordinates.
Key Point In all of these calculations, we've thought of the outside two integrals in the triple integral as giving an integral over a projected region in the xy-plane. Then the inside integral gives the sum of the integrand over the remaining variable (z) at the point (x,y) in the plane. We can also think of the inside two integrals giving an integral over a 2D slice of the volume, and the outside integral adding up all of those slices for all values of the remaining variable.
This latter way of thinking about the integral will be particularly useful when setting up integrals in Spherical Coordinates.
Spherical Coordinates are rho, theta and phi.
Game: Sketch the surface given by phi = pi/6, and that given by theta = pi/3.
Example: set up the integral of the function f(x,y,z) = xz over the volume given by the snow cone bounded above by x² + y² + z² = 8 and below by z = sqrt(x² + y²).

2009-11-02: Triple Integrals and Spherical Coordinates