# 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.

Q1:

Give an expression for the Jacobian for the change of variables from polar to Cartesian coordinates on .

• A
• B
• C
• D
• E

Q2:

Give an expression for the Jacobian for the change of variables from Cartesian to polar coordinates on .

• A
• B
• C
• D
• E

Q3:

The region is bounded by the ellipse . The transformation , , allows the integral to be expressed as an iterated double integral. Write this integral clearly indicating the limits and integrand.

• A
• B
• C
• D
• E

Q4:

Use the transformation , to evaluate , where is the triangle with vertices , , and . Give your answer to two decimal places.

Q5:

We would like to use the change of variable , to compute the integral , where is the interior of the triangle on vertices , and .

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

• A, ,
• B, ,
• C, ,
• D, ,
• E, ,

The transformed integral is . What is the integrand ?

• A
• B
• C
• D
• E

Q6:

Compute the Jacobian of the transformation of given by and .

Q7:

If is the change of coordinates from polar to Cartesian coordinates so that , then the Jacobian of this transformation is the determinant . If is a second transformation from Cartesian to another coordinate system, say , then the product of the Jacobians is the Jacobian of the composition. So, .

What does this tell us about the relation between the Jacobians and ?

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

Deduce from .

• A
• B
• C
• D
• E

Q8:

Let be the region enclosed by the ellipse . What is the image of under the transformation , ?

• AThe rectangle and
• BThe circle
• CThe circle
• DThe rectangle and
• EThe rectangle and

Q9:

Use the change of variables , to evaluate the integral , where is the region bounded by the ellipse . Give an exact answer involving .

• A
• B
• C
• D
• E

Q10:

Which of the following transformations would convert , with the region shown, into an iterated integral over a right triangle? • A,
• B,
• C,
• D,
• E,