Section 16.8

1. Section 16.8 Triple Integrals in Cylindrical and Spherical Coordinates

2. CYLINDRICAL COORDINATES Recall that Cartesian and Cylindrical coordinates are related by the formulas x = r cos θ, y = r sin θ, x2 + y2 = r2. As a result, the function f (x, y, z) transforms into f (x, y, z) = f (r cos θ,r sin θ, z) = F(r, θ, z).

3. TRIPLE INTEGRATION WITH CYLINDRICAL COORDINATES Let E be a type 1 region and suppose that its projection D in the xy-plane can be described by D = {(r, θ) | α≤θ≤β, h1(θ) ≤θ≤h2(θ)} If f is continuous, then NOTE: The dz dy dx of Cartesian coordinates becomes rdz dr dθ in cylindrical coordinates.

4. EXAMPLES 1. Find the mass of the ellipsoid E given by 4x2 + 4y2 + z2 = 16, lying above the xy-plane. Then density at a point in the solid is proportional to the distance between the point and the xy-plane. 2. Evaluate the integral

5. SPHERICAL COORDINATES AND SPHERICAL WEDGES The equations that relate spherical coordinates to Cartesian coordinates are In spherical coordinates, the counterpart of a rectangular box is a spherical wedge

6. Divide E into smaller spherical wedges Eijk by means of equally spaced spheres ρ = ρi, half-planes θ = θj, and half-cones φ = φk. Each Eijk is approximately a rectangular box with dimensions Δρ, ρiΔφ, and ρi sin φkΔθ. So, the approximate volume of Eijk is given by Then the triple integral over E can be given by the Riemann sum

7. TRIPLE INTEGRATION IN SPHERICAL COORDINATES The Riemann sum on the previous slide gives us where E is a spherical wedge given by

8. EXTENSION OF THE FORMULA The formula can be extended to included more general spherical regions such as The triple integral would become

9. EXAMPLES 1. Use spherical coordinates to evaluate the integral 2. Find the volume of the solid region E bounded by below by the cone and above by the sphere x2 + y2 + z2 = 3z.