Chapter 15 – Multiple Integrals 15.9 Triple Integrals in Spherical Coordinates Objectives: Use equations to convert rectangular coordinates to spherical coordinates Use spherical coordinates to evaluate triple integrals Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Spherical Coordinates Another useful coordinate system in three dimensions is the spherical coordinate system. It simplifies the evaluation of triple integrals over regions bounded by spheres or cones. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Spherical Coordinates The spherical coordinates (ρ, θ, Φ) of a point P in space are shown. ρ = |OP| is the distance from the origin to P. θ is the same angle as in cylindrical coordinates. Φ is the angle between the positive z-axis and the line segment OP. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Spherical Coordinates Note: ρ ≥ 0 0 ≤ θ ≤ π Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Spherical Coordinate System The spherical coordinate system is especially useful in problems where there is symmetry about a point, and the origin is placed at this point. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Sphere For example, the sphere with center the origin and radius c has the simple equation ρ = c. This is the reason for the name “spherical” coordinates. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Half-plane The graph of the equation θ = c is a vertical half-plane. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Half-cone The equation Φ = c represents a half-cone with the z-axis as its axis. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Spherical and Rectangular Coordinates The relationship between rectangular and spherical coordinates can be seen from this figure. To convert from spherical to rectangular coordinates, we use the equations x = ρ sin Φ cos θ y = ρ sin Φ sin θ z = ρ cos Φ The distance formula shows that: ρ2 = x2 + y2 + z2 Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Spherical and Rectangular Coordinates To convert from rectangular to spherical coordinates, we use the equations Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Example 1 Plot the point whose spherical coordinates are given. Then find the rectangular coordinates of the point. a) b) Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Example 2 Change from rectangular to spherical coordinates. a) b) Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Example 3 Write the equation in spherical coordinates. a) b) Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Evaluating Triple Integrals In the spherical coordinate system, the counterpart of a rectangular box is a spherical wedge where: a ≥ 0, β – α ≤ 2π, d – c ≤ π Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Visualization A region in spherical coordinates Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Evaluating Triple Integrals Although we defined triple integrals by dividing solids into small boxes, it can be shown that dividing a solid into small spherical wedges always gives the same result. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Evaluating Triple Integrals The figure shows that Eijk is approximately a rectangular box with dimensions: Δρ, ρi ΔΦ (arc of a circle with radius ρi, angle ΔΦ) ρi sinΦk Δθ (arc of a circle with radius ρi sin Φk, angle Δθ) Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Evaluating Triple Integrals Using the idea of Riemann Sum, we can write the sum as where and is some point in Eijk. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Evaluating Triple Integrals Which leads to the following integral called formula 3: where E is a spherical wedge given by: Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Spherical Coordinates Formula 3 says that we convert a triple integral from rectangular coordinates to spherical coordinates by writing: x = ρ sin Φ cos θ y = ρ sin Φ sin θ z = ρ cos Φ Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Spherical Coordinates That is done by: Using the appropriate limits of integration. Replacing dV by ρ2 sin Φ dρ dθ dΦ. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Triple Integrals in Spherical Coordinates The formula can be extended to include more general spherical regions such as: The formula is the same as in Formula 3 except that the limits of integration for ρ are g1(θ, Φ) and g2(θ, Φ). Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Triple Integrals in Spherical Coordinates Usually, spherical coordinates are used in triple integrals when surfaces such as cones and spheres form the boundary of the region of integration. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Example 4 Sketch the solid whose volume is given by the integral and evaluate the integral. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Example 5 Set up the triple integral of an arbitrary continuous function f (x, y, z) in cylindrical or spherical coordinates over the solid shown. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Example 6 Use spherical coordinates. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Example 7 Use spherical coordinates. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Example 8 Use cylindrical or spherical coordinates, whichever seems more appropriate. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Example 9 Evaluate the integral by changing to spherical coordinates. Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

More Examples The video examples below are from section 15.9 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. Example 3 Example 4 Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates

Demonstrations Feel free to explore these demonstrations below. Spherical Coordinates Exploring Spherical Coordinates Dr. Erickson 15.9 Triple Integrals in Spherical Coordinates