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

Example

We will integrate over the solid T formed by taking all nonnegative (r,theta,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 dθ dr 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, g(z) to h(z), alpha(r,z) to beta(r,z) of f(r,theta,z) r dtheta dr 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 dtheta dr 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, g(z) to h(z), alpha(r,z) to beta(r,z) of f(r,theta,z) r dtheta dr 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 dtheta dr dz with arrows to dr and middle integral sign

    The slice formed by keeping z constant is a ball. 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

    r is between 0 and sqrt(4-z).

    z slice, 2 dimensional view for range of r

    Thus the limits of the middle integral sign are 0 and sqrt(4-z).

    integral from 0 to 4, 0 to sqrt(4-z), alpha(r,z) to beta(r,z) of f(r,theta,z) r dtheta dr 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 dtheta dr dz with arrows to dr and inner integral sign

    We again view a slice formed by keeping z constant.

    z slice of paraboloid

    θ is between 0 and 2π.

    z slice, 2 dimensional view for range of theta

    Thus the final integral is as follows:

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

Other Orders of Integration

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