Lesson Worksheet: Jacobians and Change of Variables in Multiple Integrals Mathematics

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In this worksheet, we will practice making change of variables in multiple integrals and finding the Jacobian of the function of transformation of variables.

Q1:

Give an expression for the Jacobian 𝜕(𝑥,𝑦)𝜕(𝑟,𝜃) for the change of variables from polar to Cartesian coordinates on ℝ.

  • A𝑟(𝜃−𝜃)cossin
  • B𝑟
  • C𝑟
  • D𝑟(2𝜃)cos
  • Ecossin𝜃−𝜃

Q2:

Give an expression for the Jacobian 𝜕(𝑟,𝜃)𝜕(𝑥,𝑦) for the change of variables from Cartesian to polar coordinates on ℝ.

  • A1√𝑥+𝑦
  • B1𝑥𝑦
  • C1√𝑥+𝑦
  • D√𝑥+𝑦
  • E√𝑥+𝑦

Q3:

The region 𝑅 is bounded by the ellipse 9𝑥+4𝑦=7. The transformation 𝑥=𝑟3𝜃cos, 𝑦=𝑟2𝜃sin, allows the integral 𝑓(𝑥,𝑦)𝐴d to be expressed as an iterated double integral. Write this integral clearly indicating the limits and integrand.

  • A𝑓𝑟3𝜃,𝑟2𝜃𝑟𝜃𝑟√cossindd
  • B𝑓𝑟3𝜃,𝑟2𝜃16𝜃𝑟√cossindd
  • C𝑓𝑟3𝜃,𝑟2𝜃𝜃𝑟√cossindd
  • D𝑓𝑟3𝜃,𝑟2𝜃6𝑟𝜃𝑟√cossindd
  • E𝑓𝑟3𝜃,𝑟2𝜃𝑟6𝜃𝑟√cossindd

Q4:

Use the transformation 𝑥=−𝑢+𝑣, 𝑦=−13𝑢+2𝑣 to evaluate (2𝑥+𝑦)𝐴d, where 𝑅 is the triangle with vertices (0,0), (1,2), and (3,1). Give your answer to two decimal places.

Q5:

We would like to use the change of variable 𝑥=−𝑢+𝑣, 𝑦=−13𝑢+2𝑣 to compute the integral (2𝑥+𝑦)𝐴d, where 𝑅 is the interior of the triangle on vertices (0,0)(1,2), and (3,1).

What is the region 𝑆 in the 𝑢𝑣-plane over which the transformed integral is taken? Give your answer in terms of its bounding edges.

  • A𝑢=0, 𝑣=0, 𝑣=𝑢+53
  • B𝑢=0, 𝑣=0, 𝑣=𝑢3
  • C𝑢=0, 𝑣=0, 𝑣=𝑢−53−5
  • D𝑢=0, 𝑣=0, 𝑣=𝑢3+5
  • E𝑢=0, 𝑣=0, 𝑣=𝑢+33

The transformed integral is 𝐼(𝑢,𝑣)𝑢𝑣dd. What is the integrand 𝐼(𝑢,𝑣)?

  • A−359𝑢+203𝑣
  • B−73𝑢+4𝑣
  • C359𝑢+203𝑣
  • D359𝑢−203𝑣
  • E353𝑢−203𝑣

Determine (2𝑥+𝑦)𝐴d. Give your answer to 2 decimal places.

Q6:

Compute the Jacobian of the transformation of ℝ given by 𝑥=2𝑢+𝑣 and 𝑦=𝑢+2𝑣.

Q7:

If 𝑇=𝑇(𝑟,𝜃) is the change of coordinates from polar to Cartesian coordinates so that 𝑇(𝑟,𝜃)=(𝑟𝜃,𝑟𝜃)cossin, then the Jacobian of this transformation is the determinant 𝜕(𝑥,𝑦)𝜕(𝑟,𝜃). If 𝑆=𝑆(𝑥,𝑦) is a second transformation from Cartesian to another coordinate system, say (𝑠,𝑡), then the product of the Jacobians is the Jacobian of the composition. So, 𝜕(𝑠,𝑡)𝜕(𝑟,𝜃)=𝜕(𝑠,𝑡)𝜕(𝑥,𝑦)𝜕(𝑥,𝑦)𝜕(𝑟,𝜃).

What does this tell us about the relation between the Jacobians 𝜕(𝑥,𝑦)𝜕(𝑟,𝜃) and 𝜕(𝑟,𝜃)𝜕(𝑥,𝑦)?

  • AThey are the same.
  • BThey are reciprocals of each other.
  • CTheir product is zero.
  • DThey are negatives of each other.
  • ETheir product is −1.

Deduce 𝜕(𝑟,𝜃)𝜕(𝑥,𝑦) from 𝜕(𝑥,𝑦)𝜕(𝑟,𝜃).

  • A1√𝑥+𝑦
  • B√𝑥+𝑦
  • C1√𝑟+𝜃
  • D1√𝑥+𝑦
  • E𝑟

Q8:

Let 𝑅 be the region enclosed by the ellipse 9𝑥+4𝑦=10. What is the image of 𝑅 under the transformation 𝑥=𝑟3𝜃cos, 𝑦=𝑟2𝜃sin?

  • AThe rectangle 0≤𝑟≤√5 and 0≤𝜃≤𝜋
  • BThe circle 0≤𝑟≤√5
  • CThe circle 0≤𝑟≤√10
  • DThe rectangle 0≤𝑟≤√10 and 0≤𝜃≤2𝜋
  • EThe rectangle 0≤𝑟≤√5 and 0≤𝜃≤2𝜋

Q9:

Use the change of variables 𝑥=𝑟3𝜃cos, 𝑦=𝑟2𝜃sin to evaluate the integral 𝑥+𝑦𝑅d, where 𝑅 is the region bounded by the ellipse 9𝑥+4𝑦=25. Give an exact answer involving 𝜋.

  • A8,125𝜋864
  • B8,125𝜋144
  • C1,625𝜋648
  • D8,125𝜋1,296
  • E1,625𝜋108

Q10:

Which of the following transformations would convert 𝑓(𝑥,𝑦)𝑅d, with the region shown, into an iterated integral over a right triangle?

  • A𝑥=𝑢−𝑣, 𝑦=16𝑢−𝑣
  • B𝑥=𝑢−𝑣, 𝑦=−13𝑢+2𝑣
  • C𝑥=−3𝑢+35𝑣, 𝑦=−15𝑢+65𝑣
  • D𝑥=2𝑢−𝑣, 𝑦=13𝑢−𝑣
  • E𝑥=35𝑢−35𝑣, 𝑦=15𝑢−65𝑣

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