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Mathematics · lesson 4 of 4

Multiple integrals and Jacobians

Draw or describe the region before changing integration order.

University8 practice questionsNotes + worked examples
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Learn the ideas

Prerequisites: Single-variable integration and determinants.

Bounds describe a region

Draw or describe the region before changing integration order.

Area scaling

Under a smooth one-to-one change of variables, the area factor is the absolute Jacobian determinant.

Polar coordinates

x=r cos θ, y=r sin θ and dA=r dr dθ for r≥0.

Orientation versus area

A signed Jacobian tracks orientation; an area integral uses its absolute value.

Worked examples

Example 1

Integrate xy over 0≤x≤2 and 0≤y≤3.

The product of integrals is (2²/2)(3²/2)=9.

9

Example 2

For x=2u and y=3v, find the absolute Jacobian determinant.

The derivative matrix is diag(2,3); its determinant is 6.

6

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 Integrate xy over 0≤x≤2 and 0≤y≤3.

Show answer and explanation

9

The product of integrals is (2²/2)(3²/2)=9.

2 For x=2u and y=3v, find the absolute Jacobian determinant.

Show answer and explanation

6

The derivative matrix is diag(2,3); its determinant is 6.

3 What area factor accompanies polar coordinates?

Show answer and explanation

r

The magnitude of the planar polar Jacobian is r.

4 Find the area of a disk of radius 3. Enter the coefficient of π.

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9

Integrating r dr dθ gives πR²=9π.

5 Reverse the order in the integral with 0≤x≤1 and 0≤y≤x, keeping the integrand arbitrary.

Sample answer and marking guidance

The same triangular region is 0≤y≤1 and y≤x≤1, so integrate x from y to 1, then y from 0 to 1.

Credit correct limits describing the same triangle, not a rectangle.

6 Integrate x+y over the unit square.

Show answer and explanation

1

The x part contributes 1/2 and the y part contributes 1/2.

7 Explain why ignoring the Jacobian in x=2u,y=3v gives a wrong area.

Sample answer and marking guidance

A unit uv-rectangle becomes an xy-rectangle of area 6. Without the Jacobian, the transformed integral undercounts this local area by a factor of 6.

Credit a geometric scaling argument.

8 A coordinate change has determinant −5 at a point. Which factor belongs in an ordinary area integral there?

Show answer and explanation

5

Area is nonnegative, so use the absolute determinant.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.