Setting up an Integral Over a Solid with Order of Integration dr dθ dz

Example

We will integrate over the solid T formed by taking all nonnegative (r,θ,z) under the paraboloid z = 4 - x2 - y2. We convert the equation of the paraboloid to cylindrical coordinates, getting z = 4 - r2.
solid formed by taking all points under the paraboloid z = 4 - x^2 - y^2 and above xy-plane

Basic Steps with Order dr dθ dz

To determine the limits of integration, we take the outermost variable and work inward. 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
  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

    Since z is between 0 and 4, the limits of the first integral sign are 0 and 4.

    integral from 0 to 4, alpha(z) to beta(z), g(theta,z) to h(theta,z) of f(r,theta,z) r dr dtheta dz

  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

    The slice formed by keeping z constant is a ball.

    solid paraboloid with z slice

    θ is between 0 and 2π.

    z slice, 2 dimensional view for range of theta

    Thus the limits of the middle integral sign are 0 and 2π.

    integral from 0 to 4, 0 to 2pi, g(theta,z) to h(theta,z) of f(r,theta,z) r dr dtheta dz

  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

    Solving for r in the equation z = 4 - r2, we get that r = sqrt(4-z).

    z slice with equation of surface in terms of r

    Since r is between 0 and sqrt(4-z), these are the limits of the innermost integral sign.

    z slice, 2 dimensional view for range of r

    Thus the final integral is as follows:

    integral from 0 to 4, 0 to 2pi, 0 to sqrt(4-z) of f(r,theta,z) r dr dtheta dz

Other Orders of Integration

Back to Integrating Using Cylindrical Coordinates.
Back to Describing Surfaces Using Different Coordinate Systems.