A student converts ∬_R f(x,y) dx dy to polar coordinates by substituting x = r cosθ, y = r sinθ and writes ∬_S f(r cosθ, r sinθ) dr dθ. What is wrong with this?
AThey used the wrong substitution formulas for polar coordinates
BThey forgot to include the Jacobian factor r, so the integral doesn't account for how area changes with r
CThe limits of integration must be changed before any substitution is valid
DNothing — substituting coordinates and adjusting limits is all that is required
The most common error in change of variables is omitting |det(J)|. For polar coordinates, the Jacobian determinant equals r, which must be included inside the integral. Without it, the integral incorrectly treats all (r, θ) area elements as equal size — but a thin wedge at large r covers much more actual area than the same wedge near the origin. The factor r corrects for this distortion.
Question 2 Multiple Choice
Why can't you factor |det(J)| out of the integral as a constant after performing a change of variables?
AThe Jacobian is always equal to 1, so there is nothing to factor out
B|det(J)| varies across the domain as a function of the new coordinates, so it must remain inside the integral
CYou can only factor it out if the region of integration is a rectangle in the new coordinates
DFactoring the Jacobian out would change the integration variable
|det(J)| is the local area magnification factor at each point in the domain, and it generally varies from point to point. For polar coordinates, |det(J)| = r, which changes with r — pulling it outside the integral would treat all r-values as having the same magnification, which is false. Only if the transformation is globally uniform (constant Jacobian) could factoring be valid.
Question 3 True / False
In single-variable substitution u = g(x), the factor g'(x) plays a different conceptual role than |det(J)| does in multivariable change of variables.
TTrue
FFalse
Answer: False
They play exactly the same role: both are scaling factors that correct for how the substitution stretches or compresses the domain. g'(x) measures how the single-variable substitution locally magnifies the number line; |det(J)| measures how the multivariable transformation locally magnifies area or volume. The Jacobian determinant is precisely the generalization of the derivative to multiple dimensions.
Question 4 True / False
The extra factor of r that appears in polar coordinate integrals (as in ∬ f(r, θ) r dr dθ) is the absolute value of the Jacobian determinant of the polar coordinate transformation.
TTrue
FFalse
Answer: True
Computing the Jacobian matrix for x = r cosθ, y = r sinθ gives [[cosθ, −r sinθ], [sinθ, r cosθ]], whose determinant is r cos²θ + r sin²θ = r. This is why r appears in every polar integral — it's not a convention or an add-on, it's exactly the Jacobian factor required by the change-of-variables formula.
Question 5 Short Answer
Explain in your own words why you must multiply by |det(J)| when changing variables in a multivariable integral — what does this factor represent geometrically?
Think about your answer, then reveal below.
Model answer: |det(J)| measures how much the coordinate transformation locally magnifies or compresses area at each point. When you substitute new coordinates, small rectangles in the new (u,v)-space correspond to differently-sized regions in the original (x,y)-space. The integral must be corrected by this magnification factor so it accumulates the right amount of area. Integrating without |det(J)| measures the function's values but ignores how much actual area each (u,v) rectangle represents — dimensionally, the answer would be wrong.
The geometric picture is essential: you are remapping a region, and remapping distorts areas. |det(J)| is the ratio of distortion at each point, and it must be incorporated into the integrand to preserve the integral's meaning.