Change of Variables in Multiple Integrals: Notes and FAQs

Some regions are painful in x and y but beautiful in other coordinates: a disk becomes a simple rectangle in polar coordinates, and a tilted parallelogram becomes a plain square under the right linear map. Changing variables lets you rewrite an integral over a friendly region — but the area element has to be corrected, or every answer comes out wrong.

The rule is: &iint;R f(x, y) dA = &iint;S f(x(u,v), y(u,v)) · |J| du dv, where J = ∂(x, y)/∂(u, v) is the Jacobian determinant of the transformation. The factor |J| measures how much the transformation stretches area at each point. The art of the method is choosing u and v so that both the region S and the integrand become simple.

Key Concepts

• Transformation: x = x(u, v), y = y(u, v) maps a region S in the uv-plane onto R in the xy-plane.
• Jacobian: J = ∂(x, y)/∂(u, v) = xu yv − xv yu — always use its absolute value |J| in integrals.
• Polar coordinates: x = r cos θ, y = r sin θ, |J| = r, so dA = r dr dθ.
• Choose the transformation to simplify the region’s boundary, the integrand, or — ideally — both.
• Triple integrals: the same idea with a 3×3 Jacobian (cylindrical: r; spherical: ρ² sin φ).

Formulas

• &iint;R f dA = &iint;S f(u, v) |∂(x, y)/∂(u, v)| du dv
• Polar: x = r cos θ, y = r sin θ; dA = r dr dθ.
• Linear map u = ax + by, v = cx + dy: |J| is the constant 1/|ad − bc|.

Worked Example

Find the area of the disk x² + y² ≤ 4 using polar coordinates.

1. Substitute. x = r cos θ, y = r sin θ. The disk becomes the rectangle 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π.
2. Area integral. Area = &iint; r dr dθ = ∫02π ∫02 r dr dθ (integrand 1, Jacobian factor r).
3. Inner: ∫02 r dr = [r²/2]02 = 2.
4. Outer: ∫02π 2 dθ = 4π.
5. Answer: 4π square units — exactly πr² with r = 2. In x and y this integral needs messy square-root limits; in polar it is two easy steps.

Common Misconceptions

• “You just substitute x and y and keep dA.” Forgetting the |J| factor is the most common error — dA = |J| du dv, never just du dv.
• “The Jacobian is optional when the transformation looks simple.” Even for polar coordinates, dropping the r gives the wrong answer.
• “The new limits are the same numbers as the old limits.” The region itself transforms — always re-derive the limits in the new variables by mapping the boundary.

Real-World Applications

• Physics: moments of inertia of disks and spheres are far easier in polar and spherical coordinates.
• Engineering: heat flow and stress in circular or cylindrical parts.
• Probability: transforming random variables — e.g. switching to polar coordinates to handle the normal distribution.
• Computer graphics: integrating over mapped textures and parametrized surfaces.

Change of variables is a trade: you pay the Jacobian factor |J| and gain a simpler region, a simpler integrand, or both. Map the boundary to find the new limits, never forget the absolute value on the Jacobian, and let the symmetry of the problem — circles suggest polar, spheres suggest spherical — choose the coordinates for you.