Natural Science Department – Duy Tan University

1. Natural Science Department – Duy Tan University Triple Integrals in Spherical Coordinates In this section, we will learn about: Convert rectangular coordinates to spherical ones and use this to evaluate triple integrals. Lecturer: Ho Xuan Binh Da Nang-09/2014

2. 1 Spherical Coordinates Natural Science Department – Duy Tan University x = ρsin Φcos θ y = ρsin Φsin θ z = ρcos Φ ρ2 = x2 + y2 + z2 Triple Integrals in Spherical Coordinates

3. 2 Evaluating Triple InteGS. With SPH. CoorDS. Natural Science Department – Duy Tan University In the spherical coordinate system, the counterpart of a rectangular box is a spherical wedge. where: a ≥ 0, β – α≤ 2π, d – c ≤ π Triple Integrals in Spherical Coordinates

4. 3 Evaluating Triple Integrals Natural Science Department – Duy Tan University Where E is a spherical wedge given by: Triple Integrals in Spherical Coordinates

5. 4 Triple Integral In SPH. Coordinates. Natural Science Department – Duy Tan University *Writing: x = ρsin Φcos θ y = ρsin Φsin θ z = ρcos Φ • *Replacing dV by • ρ2 sin Φ dρ dθ dΦ. Triple Integrals in Spherical Coordinates

6. 5 Example 1 Natural Science Department – Duy Tan University Evaluate where B is the unit ball: Triple Integrals in Spherical Coordinates

7. 6 Example 2 Natural Science Department – Duy Tan University Use spherical coordinates to find the volume of the solid that lies: *Above the cone *Below the sphere x2 + y2 + z2 = z Triple Integrals in Spherical Coordinates

8. Thank you for your attention