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Section Change of Variables
Worksheet IC-4 Reading
Objectives
Things you should understand and know:
What it means for a transformation to be one-to-one
Determine the image of a region under a given transformation
What the Jacobian represents
Things you should be able to do:
Use a transformation to convert between coordinate systems
Find the Jacobian of a transformation
Use the Jacobian to evaluate an integral in a different coordinate system
You will be assessed on the following criteria:
Problem structure: Solution should be organized and easy to follow, with clear, complete steps that follow progressively from one to the next.
Change of Variables in Multiple Integrals: Change variables and compute multiple integrals from rectangular to polar, spherical, cylindrical, or other coordinates and back.
Graphical Interpretation of the Jacobian: Interpret the Jacobian as a scaling factor for area or volume when changing variables in multiple integrals. Be able to graph the regions in both coordinate systems.
Applied Problems: Solve applied problems using a change of variables in multiple integrals.
All of the readings for this course come from the OpenStax Calculus Volume 3 textbook, which is available for free online at
https://openstax.org/details/books/calculus-volume-3 . You can read the textbook online or download a PDF version. The textbook is also available in print from various retailers if you prefer a physical copy.
For this lesson, you should read sections
5.7 Change of Variables in Multiple Integrals . You should read through the sections carefully, making sure to understand the definitions, examples, and key ideas. You should also work through the exercises at the end of each section to test your understanding and practice applying the concepts. Some of the exercises at the end of the section are assigned in the WP assignments.
Worksheet IC-4 Videos
Change of Variables in Multiple Integrals
Worksheet IC-4 Written Practice
INFORMATION ABOUT ALL WRITTEN PRACTICE (WP) ASSIGNMENTS:
THE WP ASSIGNMENTS ARE DESIGNED TO PREPARE YOU FOR THE QUIZZES AND ASSESSMENTS
Work on every problem on every assignment.
If you get stuck or do not understand a solution, ask questions.
For each problem, try it once or twice before looking at solutions or asking for help.
Mark the problems you did correctly on the first try.
Highlight problems where you got help (using solutions, videos, a tutor, etc) -These are the problem types that you will need more practice on to be ready for the assessments.
Understand that just copying the solutions instead of working through the problems will greatly reduce your chances of success in this class.
Change of Variables in Multiple Integrals: Complete problems 381, 392, 393, 397
In addition to these problems, upload a link to a solution video: Pick one problem from Change of Variables problems 378-397 (the problems should not be any listed above). Create a short video of you presenting the solution to your classmates. You may use any video software (zoom, flip, explain everything, etc). The video does not need to be perfect or fully edited (mistakes are welcome), but it should show your thought process as you go through the solution.
Worksheet IC-4 Sample Outcomes
For the quiz on this outcome you would be given 30 minutes to complete 2 problems. Here are samples of the types of problems you will encounter:
1. Given
\(\int_0^4\int_{y/2}^{y/2+1} x-\frac{y}{2}dxdy\) and the transformation
\(T:(u,v)\rightarrow(x,y)\) by
\(T(u,v)=(u+v,2v)\)
(a) Sketch the region of integration in the
\(xy\) -plane.
Answer .
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(b) Sketch the region of integration in the
\(uv\) -plane.
Answer .
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(c) Find the Jacobian of the transformation.
Hint .
To find the Jacobian, compute the partial derivatives of
\(x\) and
\(y\) with respect to
\(u\) and
\(v\text{.}\)
Answer . Solution .
\(J(u,v) = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}\)
\(J(u,v) = \begin{vmatrix} 1 & 1 \\ 0 & 2 \end{vmatrix}\)
(d) Use the transformation and the Jacobian to evaluate the integral.
Hint .
Substitute the transformation into the integral and multiply by the Jacobian.
Answer .
\(\int_0^4\int_{y/2}^{y/2+1} x-\frac{y}{2}dxdy = 2\)
Solution .
Using the region in the
\(uv\) -plane, we have:
\(\int_0^4\int_{y/2}^{y/2+1} x-\frac{y}{2}dxdy =\int_0^2\int_0^1 (u+v)-v dv du\)
\(= \int_0^2\int_0^1 u dv du\)
\(= \left[\frac{u^2}{2}\right]_0^2\)
2. Given
\(x=e^{2u-v}\) and
\(y=e^{u+v}\text{.}\)
(a) Find the Jacobian,
\(J(u,v)\text{.}\)
Hint .
To find the Jacobian, compute the partial derivatives of
\(x\) and
\(y\) with respect to
\(u\) and
\(v\text{.}\)
Answer . Solution .
To find the Jacobian, we compute the partial derivatives:
\(\frac{\partial x}{\partial u} = 2e^{2u-v}, \quad \frac{\partial x}{\partial v} = -e^{2u-v}\)
\(\frac{\partial y}{\partial u} = e^{u+v}, \quad \frac{\partial y}{\partial v} = e^{u+v}\)
\(J(u,v) = \begin{vmatrix} 2e^{2u-v} & -e^{2u-v} \\ e^{u+v} & e^{u+v} \end{vmatrix}\)
\(= 2e^{2u-v} \cdot e^{u+v} - (-e^{2u-v}) \cdot e^{u+v}\)
(b) Use the above Jacobian to evaluate
\(\iint_R (xy) dA\) where
\(R\) is the triangle defined by the equations
\(y=x, x=1\text{,}\) and
\(y=2x\text{.}\)
Hint . Answer .
\(\iint_R (xy) dA = \frac{7e^{6}}{24}\)
Solution .
First we find the region in the
\(uv\) -plane that corresponds to
\(R\text{.}\)
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Next we find the region in the
\(uv\) -plane that corresponds to
\(R\text{.}\)
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Now we can evaluate the integral:
\(\iint_R (xy) dA = \iint_R e^{3u} du dv\)
\(\iint_R (xy) dA = \int_{1}^{2} \int_{2-ln(2)}^{2} e^{3u} du dv\)
\(\iint_R (xy) dA = \int_{1}^{2} \left[\frac{e^{3u}}{3}\right]_{2-ln(2)}^{2} dv\)
\(\iint_R (xy) dA = \int_{1}^{2} \frac{e^{6}-e^{6-3ln(2)}}{3} dv\)
\(\iint_R (xy) dA = \int_{1}^{2} \frac{e^{6}-\frac{e^{6}}{8}}{3} dv\)
\(\iint_R (xy) dA = \int_{1}^{2} \frac{7e^{6}}{24} dv\)
\(\iint_R (xy) dA = \left[\frac{7e^{6}}{24}v\right]_{1}^{2}\)
\(\iint_R (xy) dA = \frac{7e^{6}}{24}\)
