# Multiple integrals and Jacobians

> Draw or describe the region before changing integration order.

Course: [University calculus and multivariable methods](https://studybrix.com/en/courses/university-calculus/index.md)
Level: university
Prerequisites: Single-variable integration and determinants.

## Notes

### 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.

## Practice questions

### 1. Integrate xy over 0≤x≤2 and 0≤y≤3.

Answer: 9

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

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

Answer: 6

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

### 3. What area factor accompanies polar coordinates?

A. 1/r
B. 1
C. r²
D. r

Answer: r

Explanation: The magnitude of the planar polar Jacobian is r.

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

Answer: 9

Explanation: 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 / 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.

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

### 6. Integrate x+y over the unit square.

Answer: 1

Explanation: 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 / 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.

Explanation: Credit a geometric scaling argument.

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

A. 1/5
B. 5
C. −5
D. 0

Answer: 5

Explanation: Area is nonnegative, so use the absolute determinant.

## Further reading

- https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/
- https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.