Section 15.7

Spherical Coordinates

The natural choice for spheres, cones, and planets.

1

Introduction

Spherical coordinates use a distance from origin () and two angles ().

  • Rho (): Distance from origin ().
  • Theta (): Same angle as polar/cylindrical (0 to ).
  • Phi (): Angle from positive z-axis (0 to ). "North Pole" is 0, "South Pole" is .
2

Integration Formula

Volume Element

The "Spherical Wedge" volume factor is crucial.

Memorize this: .

3

Visualizing an "Ice Cream Cone"

Interactive: Cone in Sphere

4

Worked Examples

Example 1: Volume of a Sphere

Use spherical coordinates to prove .

1. Bounds:

Full Sphere: , (Pole to Pole), .

2. Integral:

.

3. Separate:

  • .
  • .
  • .

Product: .

Example 2: The "Ice Cream Cone"

Find volume lying above cone and inside sphere .

Wait, let's simplify for clarity:

Inside sphere centered at origin above cone .

1. Bounds:

Cone angle: .

So .

Sphere is . .

2. Integral:

.

3. Solve:

.
.
from theta.

Result: .

Example 3: Spherical Shell Mass

Find mass of region between spheres and in the upper hemisphere () with density .

1. Bounds:

Upper hemi: .

Radii: .

2. Function:

Density .

3. Integral:

.

4. Execute:

Rho: .

Phi: .

Theta: .

Mass: .

5

Practice Quiz

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