Integrating in Spherical Coordinates

When estimating an integral using spherical coordinates we divide up a solid into small pieces of dimension Δρ by Δθ by Δφ.

solid formed by taking Delta rho by Delta theta by Delta phi

The shaded portion has volume Δθ((ρ+Δρ)3-ρ3)(cosφ-cos(φ+Δφ))/3 = Δθ(3ρ2Δρ+3ρ(Δρ)2+(Δρ)3)(cosφ-cosφcos(Δφ)+sinφsin(Δφ))/3 since cos(a+b)=cos(a)cos(b)-sin(a)sin(b). When Δρ, Δθ, and Δφ are small, the volume is approximately ρ2 sinφ Δρ Δθ Δφ since cos(Δφ) is approximately 1 and sin(Δφ) is approximately Δφ. This is a visual aid for why adding an extra ρ2 sinφ is necessary when using spherical coordinates. However, the precise reason is that the absolute value of the Jacobian of the transformation x = ρ sinφ cosθ, y = ρ sinφ sinθ, z = ρ cosφ, is ρ2 sinφ. Therefore, when we integrate a function f(ρ,θ,φ) over a solid T in spherical coodinates, we use

triple integral over T of f(rho,theta,phi) rho^2 sin(phi) drho dtheta dphi

We can also change the order of integration if T is a basic solid (the boundary is a finite number of continuous surfaces--see Calculus One and Several Variables 8th Ed. Salas/Hille/Etgen pg 999.) Thus the following integrals are all equivalent:

triple integral over T of f(rho,theta,phi) rho^2 sin(phi) drho dphi dtheta triple integral over T of f(rho,theta,phi) rho^2 sin(phi) dtheta drho dphi triple integral over T of f(rho,theta,phi) rho^2 sin(phi) dtheta dphi drho triple integral over T of f(rho,theta,phi) rho^2 sin(phi) dphi drho dtheta triple integral over T of f(rho,theta,phi) rho^2 sin(phi) dphi dtheta drho

Basic Steps in Determining the Limits of Integration

To determine the limits of integration, we take the outermost variable and work inward. Sketching the graph of the solid T is extremely helpful.
Here are the basic step for integrating in the order dρ dθ dφ. Other orders are similar.

  1. Determine the maximum and minimum values of the outermost variable. These will be the limits of integration on the first integral sign.
    triple integral over T of f(rho,theta,phi) rho^2 sin(phi) drho dtheta dphi with arrows to dphi and outer integral sign

  2. View a slice formed by keeping the outermost variable constant. Now determine the maximum and minimum values of the middle variable within that slice in terms of the outermost variable. This will give the limits of integration for the middle integral. Note that if the maximum and minimum values depend on where the slice is taken, you will need to split the integral.
    triple integral over T of f(rho,theta,phi) rho^2 sin(phi) drho dtheta dphi with arrows to dtheta and middle integral sign

  3. Finally, using the same slice, determine the range of the innermost variable in terms of the other two variables. This will give the limits of integration for the inner integral. Note that if the range of the innermost variable changes within the slice, you will need to split the integral.
    triple integral over T of f(rho,theta,phi) rho^2 sin(phi) drho dtheta dphi with arrows to drho and inner integral sign

The integral will have the general form

integral from alpha to beta, gamma(phi) to lambda(phi), g(theta,phi) to h(theta,phi) of f(rho,theta,phi) rho^2 sin(phi) drho dtheta dphi

Example

We will integrate over the solid T formed by taking a ball of radius 1 and intersecting it with a cylinder of radius 1/sqrt(2). Since 1/sqrt(2)=sin(π/4), at φ=π/4 the surface of the solid changes general shape.
solid formed by the intersection of a ball of radius 1 and a cylinder of radius 1/sqrt(2)

Orders of Integration for Example Solid

Back to Describing Surfaces Using Different Coordinate Systems.