Integrating in Cylindrical Coordinates

When estimating an integral using cylindrical coordinates we divide up a solid into small pieces of dimension Δr by Δθ by Δz.

solid formed by taking Delta r by Delta theta by Delta z

The shaded portion has volume [π(r+Δr)2-π r2]Δz Δθ/(2π)=rΔrΔθΔz+0.5(Δr)2ΔθΔz, which is approximately rΔrΔθΔz when Δr, Δθ, and Δz are small. This is a visual aid for why adding an extra r is necessary when integrating with cylindrical coordinates. However, the precise reason is that the absolute value of the Jacobian of the transformation x = r cosθ, y = r sinθ, z = z, is r. Therefore, when we integrate a function f(r,θ,z) over a solid T in cylindrical coodinates, we use

triple integral over T of f(r,theta,z) r dr dtheta dz

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(r,theta,z) r dr dz dtheta triple integral over T of f(r,theta,z) r dtheta dr dz triple integral over T of f(r,theta,z) r dtheta dz dr triple integral over T of f(r,theta,z) r dz dr dtheta triple integral over T of f(r,theta,z) r dz dtheta dr

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 dr dθ dz. 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(r,theta,z) r dr dtheta dz with arrows to dz 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(r,theta,z) r dr dtheta dz 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(r,theta,z) r dr dtheta dz with arrows to dr and inner integral sign

The integral will have the general form

integral from a to b, alpha(z) to beta(z), g(theta,z) to h(theta,z) of f(r,theta,z) r dr dtheta dz

Example

We will integrate over the solid T formed by taking all nonnegative (r,θ,z) under the paraboloid z=4-x2-y2.
solid formed by taking all points under the paraboloid z=4-x^2-y^2 and above xy-plane

Orders of Integration for Example Solid

Back to Describing Surfaces Using Different Coordinate Systems.