Triple Integral in Spherical Coordinates

Spherical Coordinates 𝜌 = distance |OP| (𝜌≥0) 𝜃 = angle of the projection of the point into the 𝑥𝑦-plane measured from the positive x-axis (0≤𝜃≤2𝜋 ) ϕ = angle |OP| makes with the positive z- axis (0 ≤ ϕ ≤ π; if ϕ > π/2 the point has a negative z-coordinate) Relationship with cartesian: with thus Also:

Spherical Coordinates Basic graphs in spherical coordinates: 𝜌=𝑐 represents a sphere of radius 𝑐 ( 𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑐 2 in cartesian) ϕ = c represents a cone 𝜃=𝑐 represents a vertical plane

Spherical Coordinates Plot the point and convert to cartesian: Change from rectangular to spherical: The projection of the point in the 𝑥𝑦-plane is in the first quadrant

Spherical Coordinates Example 1 Write the equation of the cone 𝑧 2 = 𝑥 2 + 𝑦 2 in spherical coordinates. Recall that Also, 𝑥 2 + 𝑦 2 = 𝑟 2 and Substituting into the equation of the cone, yields: Simplifying: Thus is the equation of the top half of the cone. is the equation of the bottom half of the cone.

Triple Integrals in spherical coordinates The volume element is Theorem: Change of coordinates where E is the spherical wedge given by

Triple Integrals in Spherical Coordinates - Example 4 Evaluate where E is the hemispherical region that lies above the 𝑥𝑦-plane and below the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 =9 Spherical coordinates: hemisphere

Triple Integrals in Spherical Coordinates - Example 5 Find the volume of the solid that lies within the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 =49, above the 𝑥𝑦-plane and outside the cone 𝑧=4 𝑥 2 + 𝑦 2 We need to determine the angle that describes the cone in spherical coordinates.

Triple Integrals in Spherical Coordinates - Example 6 Evaluate the integral by changing to spherical coordinates The solid is bounded below by the cone 𝑧= 𝑥 2 + 𝑦 2 and above by the hemisphere 𝑧= 2− 𝑥 2 − 𝑦 2 . The radius of the hemisphere is 2 The projection of the solid in the 𝑥𝑦-plane is the quarter of the disk 𝑥 2 + 𝑦 2 ≤1 in the first quadrant.

Triple Integrals in Spherical Coordinates Example 6 continued the cone has equation the sphere has equation 𝜌= 2 The limits of integration are