Change of variables
Objective. Use Jacobians in polar and general transformations.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Change of variables in multiple integrals transforms coordinates (x, y) to new coordinates (u, v) using the Jacobian determinant to correctly scale area elements: dA = |∂(x,y)/∂(u,v)| du dv. In polar coordinates x = r cos θ, y = r sin θ, the Jacobian is r, so dA = r dr dθ. This is essential when the region or integrand has circular symmetry, since Cartesian limits would be awkward. For example, ∬ e^{−(x²+y²)} dA over the plane becomes ∫₀^{2π} ∫₀^∞ e^{−r²} r dr dθ, which evaluates cleanly using substitution. Choosing coordinates matching the region's symmetry greatly simplifies computation.
Worked example
Use change of variables to polar coordinates to evaluate ∬_D (x² + y²) dA where D is the disc of radius 2.
- 1Substitute x = r cos θ, y = r sin θ so x² + y² = r², with Jacobian r.
- 2Rewrite the integral as ∫₀^{2π} ∫₀² r² · r dr dθ.
- 3Integrate over r: ∫₀² r³ dr = [r⁴/4]₀² = 4.
- 4Integrate over θ: ∫₀^{2π} 4 dθ = 8π.
Answer: 8π
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Students forget to multiply by the Jacobian factor r when converting to polar coordinates, treating dA = dr dθ instead of r dr dθ, which produces an incorrect integral value.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.