Triple integrals in spherical coordinates

Triple integrals in spherical coordinates

Lessons

Notes:

Spherical Coordinates

There are times where instead of Cartesian coordinates, we use spherical coordinates for triple integrals. For spherical coordinates, instead of x,y,zx,y,z's, we have ρ,θ,φ\rho, \theta, \varphi's. In other words,

(x,y,z)(ρ,θ,φ)(x,y,z) \to (\rho, \theta, \varphi )

spherical coordinates

Equations to Convert From Cartesian to Spherical

From the graph, we can obtain the following equations which will be useful for converting spherical to cartesian, or vice versa:

r=ρsinφr = \rho \sin \varphi
z=ρcosφz = \rho \cos \varphi
z2+r2=ρ2z^2 + r^2 = \rho^2

Using the equations from past sections, we can also obtain more equations:

x=ρsinφcosθx = \rho \sin \varphi \cos \theta
y=ρsinφsinθy = \rho \sin \varphi \sin \theta
ρ2=x2+y2+z2\rho^2 = x^2 + y^2 + z^2


Triple Integrals in Spherical Coordinates

Suppose we want to convert a triple integral of f(x,y,z)f(x,y,z) in Cartesian Coordinates to spherical Coordinates on region EE. Let region EE be:

aρb a \leq \rho \leq b
αθβ\alpha \leq \theta \leq \beta
δφγ\delta \leq \varphi \leq \gamma

Then the conversion would be:

Ef(x,y,z)dV=δγαβabf(ρsinφcosθ,ρsinφsinθ,ρcosφ)ρ2sinφdpdθdφ\int \int \int_E f(x,y,z)dV = \int_\delta^\gamma \int_\alpha^\beta \int_a^b f(\rho \sin \varphi \cos \theta, \rho \sin \varphi \sin \theta, \rho \cos \varphi)\rho^2 \sin \varphi d p d\theta d\varphi

  • Introduction
    Triple Integrals in Spherical Coordinates Overview:
    a)
    Spherical Coordinates
    • (x,y,z)(ρ,θ,φ)(x,y,z) \to (\rho, \theta, \varphi )
    • Graph of the coordinates in 3D

    b)
    Equations to Convert from Cartesian to Spherical
    • Pythagoras
    • Trig Ratios

    c)
    Triple Integrals in Spherical Coordinates
    • Region EE
    • Add an extra ρ2sinφ\rho^2 \sin \varphi
    • An Example