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Math 2263: Change of Variables Practice March 9, 2012Multivariable Calculus

Using a change of variables to compute a double or triple integral can often greatly reduce

the amount of work you have to do when integrating, but when we venture too far from thecommon coordinate changes (polar, cylindrical, spherical) it is also one of the more concep-tually difficult topics we cover in Multivariable Calculus. This worksheet aims to clear up theconfusion slightly by guiding you step by step through one particular Change of Variablesproblem. (Based on Exercise 15.10 #19 in the text.)

Remark: The arithmetic in some of the following parts will involve logarithms. You may ndit advantageous to review some common logarithm laws (Wikipedia: http://en.wikipedia.org/wiki/List_of_logarithmic_identities#Using_simpler_operations ).

(A) Let R be the region in the rst quadrant bounded by the lines y = x and y = 3 x , andthe hyperbolas xy = 1 and xy = 3. Sketch the region R . Label each boundary curveand each intersection point of two boundary curves. Express R in a way that wouldallow you to double integrate a function on R .

http://en.wikipedia.org/wiki/List_of_logarithmic_identities#Using_simpler_operationshttp://en.wikipedia.org/wiki/List_of_logarithmic_identities#Using_simpler_operationshttp://en.wikipedia.org/wiki/List_of_logarithmic_identities#Using_simpler_operationshttp://en.wikipedia.org/wiki/List_of_logarithmic_identities#Using_simpler_operations

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(B) Using part (A), compute the double integral R

xy dA .

(Hint: You will need to break R into two pieces and express the double integral as thesum of two iterated integrals.)

This integral is a little fussy and long, but not terrible. However, it is easy to see that evena minor change can make the integral much worse (if we had used xy = 4 instead of xy = 3,we would have to break R into three pieces in order to integrate). We will now computethis integral again, but this time we will use a change of variables to transform the region of integration into a much nicer shape.

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(C) Consider the transformation T given by the equations

x =uv

and y = v.

Find a region S in the uv -plane which is mapped to R under the transformation T .(Hint: Try expressing each of the boundary curves of R in terms of u and v.)Sketch S .

(D) If f (x, y ) = xy , write f in terms of u and v.

(E) Compute the Jacobian (x, y ) (u, v )

of the transformation T .

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(F) Using the Jacobi Theorem (Theorem 9 in section 15.10) and your answers to parts (C),(D), and (E), compute R xy dA by changing variables to u and v according to T .

If you did not make mistakes, your answers to (B) and (F) should be the same. Which waseasier, (A) and (B), or (C)-(F)? What if instead of two pieces, the integral in (B) had tobe broken into three or four pieces, or even more? Hopefully this illustrates how a cleverchange of variables can change a not-so-nice region of integration into a more approachableone, and make some double and triple integrals easy to compute.