1. Sec 16.7 Triple Integrals in Cylindrical Coordinates In the cylindrical coordinate system, a point P is represented by the ordered triple (r, θ, z), where r and θ are the polar coordinates of the projection of P onto the xy-plane, and z is the directed distance from the xy-plane. To convert from cylindrical to rectangular coordinates: use x = r cos θ y = r sin θ z = z To convert from rectangular to cylindrical coordinates: use Cylindrical coordinates are useful in problems with symmetry about an axis. The axis of symmetry is taken to be the z-axis.

2. Evaluating Triple Integrals with Cylindrical Coordinates: Suppose E is a Type I region, that is where D is the projection of E onto the xy-plane, and D is given in polar coordinates by D = { (r, θ) : α ≤ θ ≤ β, h 1 (θ) ≤ r ≤ h 2 (θ) }, then becomes

3. Sec 16.8 Triple Integrals in Spherical Coordinates In the spherical coordinate system, a point P is represented by the ordered triple (ρ, θ, φ), where ρ is the distance from the origin to the point P, θ is the same as in cylindrical coordinates, and φ is the angle between the positive z-axis and the line segment connecting the origin to the point P. Note that ρ ≥ 0 and 0 ≤ φ ≤ π. To convert from spherical to rectangular coordinates: use x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ To convert from rectangular to spherical coordinates: use The spherical coordinate system is useful in problems with point symmetry. The point of symmetry is taken as the origin.

4. Evaluating Triple Integrals with Spherical Coordinates: In the spherical coordinate system, the counterpart of a rectangular box is a spherical wedge.: E = { (ρ, θ, φ) : a ≤ ρ ≤ b, α ≤ θ ≤ β, c ≤ φ ≤ d }, where α ≥ 0, β – α ≤ 2π, and d – c ≤ π. In spherical coordinates,

5. Sec 16.9 Change of Variables in Multiple Integrals Sometimes an integral can be simplified by a change of variables (for example, changing rectangular to polar coordinates). Definition: The Jacobian of the transformation T given by x = g(u, v) and y = h(u, v) is

6. Change of Variables in a Double Integral To make a change of variables, the Jacobian is used: NOTE: R is the plane region as described on the xy-plane and S is the plane region as described on the uv-plane. The usual reason for a change of variables is to transform a “messy” xy-region into a nice (usually, rectangular) uv-region.