Question

Calc III Tutorial: Integration in Spherical Coordinates

Answer 1

\({\bf{Spherical~Coordinates~Defined:}}\)
consists of three coordinates (ρ, Φ, θ)
- ρ is the distance from the origin to the point (unlike r from cylindrical coordinate, ρ also incorporates distance above the xy plane)
- Φ is the angle that the point makes with the z-axis
- θ is the same as cylindrical (see cylindrical coordinates tutorial)



I only remember these from quant mech and I hated them whee

Answer 2

\({\bf{Rectangular~to~Spherical:}}\)
r = ρsinΦ
z = ρcosΦ

if you have trouble remembering these try to think of the quadrant between the z-axis, the origin, and Q

Answer 3

but honestly just remember them cause this is a major pain

Answer 4

following up from that

x = rcosθ = ρsinΦcosθ
y = rsinθ = ρsinΦsinθ

ρ = sqrt(x^2+y^2+z^2) basically like r but in three dimensions instead of 2
= sqrt(r^2 + z^2)

Answer 5

Special cases:
ρ = constant gives a sphere w/ radius equal to that constant
θ = constant gives a half-plane from the origin w/ z (I'm going to draw this)
Φ = constant gives a cone with the angle between the slanted surface and the z-axis equal to that constant (again will draw this)

Answer 6

\({\bf{Spherical~Integration:}}\)
\[\int\limits_{θ1}^{θ2}\int\limits_{Φ1}^{Φ2}\int\limits_{ρ1}^{ρ2}fρ^{2}\sinΦ~dρ~dΦ~dθ\]

Method of finding limits:
1. project onto the xy plane (see cylindrical tutorial)
2. draw a line from the origin upwards through the shape to find the ρ limits
2. draw a line from the origin through the xy projection to find the Φ limits
3. find the min and max θ that contain the shape to find the θlimits

Answer 7

theta limits work the same as cylindrical

Answer 8

Source material is section 15.7 of Thomas' Calculus, Early Transcendentals, 14th edition by Hass, Heil, Weir, et. al.