Calc 3 Lecture NotesSection 13.8Page 1 of 11

Section 13.8: Change of Variables in Multiple Integrals

Big idea: Changing variables can simplify the integration boundaries and integrands of many multiple integrals. In this section, we learn a general formalism for transforming integrals using transformations besides cylindrical or spherical.

Big skill: You should be able to find a transformation that simplifies a multiple integral.

Note the following integral transformation (from section 13.3):

Three aspects of the integral had to be transformed:

  1. The integrand was transformed using the transformations and .
  2. The integration region was transformed from a quarter-circle in the x-y plane to a rectangle in the “r-” plane:
  1. The differential area element was transformed from to by geometrically analyzing small area elements in the x-y plane. Partitions of the x-y plane had the shape of annular sectors, while corresponding partitions of the r- plane are rectangular:

In this section, we will be responsible for finding a variable transformation that simplifies the integrand and/or the region of integration. The book provides a formula for transforming the differential area or volume element once the transformation has been chosen.

A little formalism before we look at making some transformations:

A transformationT from the u-v plane to the x-y plane is a function that maps points in the u-v plane to points in the x-y plane. The shorthand notation is:

where

and

for some functions g and h.

  • A change of variables for a double integral is defined by a transformation T from a region S in the u-v plane to a region R in the x-y plane.
  • R is called the image of S under the transformation T.

  • T is one-to-one on S if for every point (x, y) in R there is exactly one point (u, v) in S such that T(u, v) = (x, y).
  • This implies we can solve for u and v in terms of x and y.
  • Also, we will restrict our transformations to those where g and h have continuous first partial derivatives in S.

Practice:

  1. Let R be the region bounded by the lines,,, and . Find a transformation T that maps a rectangular region in the u-v plane onto this parallelogram. Notice that you can show that the intersection points are (0, 0), (4/3, 8/3), (4, 4), and (8/3, 4/3). Show how the boundaries of a double integral are simplified by the transformation.

  1. Let R be the region bounded by the hyperbolae andand the lines and y = 2. Find a transformation T that maps a rectangular region in the u-v plane.Show how the boundaries of a double integral are simplified by the transformation.

Now that we’ve had some practice finding transformations, let’s derive a formula for computing the differential area element given a transformation.

First suppose that we have a transformation from (u, v) onto (x, y). Under this transformation, rectangular partitions of the region S will transform to non-rectangular partitions of the region R.

The problem is, we need the areas Ai of each of the curvilinear regions Ri, because those areas are used in computing double integrals:

The trick is to approximate each Ai as a parallelogram whose four corners come from the transformed coordinates of the rectilinear regions Si:

The points A, B, C, and D have coordinated determined by the transformation T:

These coordinates can be used to compute the vectors and . We want these vectors because the area of the parallelogram they describe can be computed from .

From the definition of partial derivatives, .

So, for u and v small,

So, our vectors simplify to:

And now we can compute the cross section:

And now we can compute area:

Definition 8.1: Jacobian of a Transformation

The determinant

is called the Jacobian of a transformation T and is written using the notation

Given this area transformation and definition, we can convert our integral to the u-v plane:

Theorem 8.1: Change of Variables in Double Integrals

If a region Sin the u-v plane is mapped onto the region R in the x-y plane by the one-to-one transformation T defined by and , where g and h have continuous first derivatives on S, and if f is continuous on R and the Jacobian is nonzero on S, then

.

Practice:

  1. Show that the Jacobian yields the correct differential area element dA for polar coordinates.
  1. Compute the Jacobian for the hyperbolic transformation from page 4.

  1. Compute for the region R shown below first using Cartesian coordinates and then using transformed coordinates.

Change of variables for triple integrals:

In three dimensions, a change of variables is fairly analogous to the two dimensional case:

Given a transformation T from a region S of u-v-w space onto a region R of x-y-z space, specified by the functions

, , and ,

the Jacobian is defined as:

Theorem 8.2: Change of Variables in Triple Integrals

If a region Sin u-v-w space is mapped onto the region R in x-y-zspace by the one-to-one transformation T defined by , , and , where g,h, and have continuous first derivatives on S, and if f is continuous on R and the Jacobian is nonzero on S, then

.

Practice:

  1. Show that the Jacobian yields the correct differential volume element dV for spherical coordinates.

  1. Toroidal coordinates are used to specify the location of points inside toroids as shown below. The center of the toroid is a circle of radius a (called the major radius) in the x-y plane, and points are located by an angle  measured from the standard position in the x-y plane, a distance r measured from the major circle, and an angle  measured in the plane  = k. The transformation is specified by:

Compute the volume of a torus with a major radius of 2 and a minor radius of 1. Compare this to the answer you get from Pappus’ Second Theorem, which says that the volume of a solid of revolution equals the cross-sectional area of the rotated lamina times the distance the traveled by its centroid.