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

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


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

  • A𝑟(𝜃𝜃)cossin
  • B𝑟
  • C𝑟
  • D𝑟(2𝜃)cos
  • Ecossin𝜃𝜃


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

  • A1𝑥+𝑦
  • B1𝑥𝑦
  • C1𝑥+𝑦
  • D𝑥+𝑦
  • E𝑥+𝑦


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,𝑟2𝜃𝑟𝜃𝑟cossindd
  • B𝑓𝑟3𝜃,𝑟2𝜃16𝜃𝑟cossindd
  • C𝑓𝑟3𝜃,𝑟2𝜃𝜃𝑟cossindd
  • D𝑓𝑟3𝜃,𝑟2𝜃6𝑟𝜃𝑟cossindd
  • E𝑓𝑟3𝜃,𝑟2𝜃𝑟6𝜃𝑟cossindd


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.


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=𝑢535
  • 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)?

  • A359𝑢+203𝑣
  • B73𝑢+4𝑣
  • C359𝑢+203𝑣
  • D359𝑢203𝑣
  • E353𝑢203𝑣

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


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


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).

  • A1𝑥+𝑦
  • B𝑥+𝑦
  • C1𝑟+𝜃
  • D1𝑥+𝑦
  • E𝑟


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 0𝑟5 and 0𝜃𝜋
  • BThe circle 0𝑟5
  • CThe circle 0𝑟10
  • DThe rectangle 0𝑟10 and 0𝜃2𝜋
  • EThe rectangle 0𝑟5 and 0𝜃2𝜋


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


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

  • A𝑥=𝑢𝑣, 𝑦=16𝑢𝑣
  • B𝑥=𝑢𝑣, 𝑦=13𝑢+2𝑣
  • C𝑥=3𝑢+35𝑣, 𝑦=15𝑢+65𝑣
  • D𝑥=2𝑢𝑣, 𝑦=13𝑢𝑣
  • E𝑥=35𝑢35𝑣, 𝑦=15𝑢65𝑣

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