16 MULTIPLE INTEGRALS

16.9 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS
SUGGESTED TIME AND EMPHASIS
1–1 1 2
classes Optional material (essential if Chapter 17 is to be covered)
POINTS TO STRESS
1. Reason for change of variables: to reduce a complicated multiple integration problem to a simpler integral
or an integral over a simpler region in the new variables
2. What happens to area over a change in variables: The role of the Jacobian
3. Various methods to construct a change of variables
∂ (x, y)
∂ (u,v)
QUIZ QUESTIONS
• Text Question: When we convert a double integral from rectangular coordinates to spherical coordinates,
where does the ρ 2 sin ϕ term come from?
Answer: It is the magnitude of the Jacobian.
• Drill Question: The unit square is the square with side length 1 and lower left corner at the origin. What
is the area of the image R (in the xy-plane) of the unit square S (in the uv-plane) under the transformation
x = u + 2v, y =−6u − v?
Answer: 11
MATERIALS FOR LECTURE
• One good way to begin this section is to discuss u-substitution from a geometric point of view. For
example, ∫ ( sin 2 x ) cos xdx is a somewhat complicated integral in x-space, but using the change of
coordinate u = sin x reduces it to the simpler integral ∫ u 2 du in u-space. If the students are concurrently
taking physics or chemistry, discuss how the semi-logarithmic paper that they use is an example of this
type of coordinate transformation.
• Note that it is very important that we take the absolute value of the Jacobian determinant. For example,
point out that the Jacobian determinant for spherical coordinates is always negative (see Example 4).
Another example of a negative Jacobian is the transformation x = u + 2v, y = 3u + v, which takes 〈1, 0〉
to 〈1, 3〉 and 〈0, 1〉 to 〈2, 1〉.
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