Multiple Integrals & Change of Variables
Key ideas for double and triple integrals, polar/cylindrical/spherical coordinates, and the Jacobian.
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Questions Covered in This Set
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What does ∬_R f(x,y) dA represent when f ≥ 0?
The volume under the surface z = f(x,y) above the region R. With f = 1 it gives the area of R; with f a density it gives total mass.
State Fubini's theorem in practical terms.
A double integral can be computed as an iterated integral: integrate with respect to one variable treating the other as constant, then integrate the result. Over a rectangle the order doesn't matter.
Rule of thumb for limits on a non-rectangular region
The outer limits must be plain numbers; the inner limits may depend on the outer variable, never the reverse. Always sketch the region first.
Why is ∫₀¹∫ₓ¹ e^{y²} dy dx impossible as written, and how do you fix it?
e^{y²} has no elementary antiderivative. Swap the order over the triangle 0 ≤ x ≤ y ≤ 1: ∫₀¹∫₀^y e^{y²} dx dy = ∫₀¹ y e^{y²} dy = ½(e − 1).
Polar area element and why the extra r?
dA = r dr dθ. A polar patch has sides dr and arc length r dθ, so its area is r dr dθ — patches farther from the origin are fatter.
Evaluate the volume under z = 4 − x² − y² over the disk of radius 2.
∫₀^{2π}∫₀² (4 − r²) r dr dθ = 2π[2r² − r⁴/4]₀² = 2π(8 − 4) = 8π.
How does polar coordinates evaluate the Gaussian integral?
I = ∫_{−∞}^{∞} e^{−x²}dx ⇒ I² = ∬_{ℝ²} e^{−(x²+y²)}dA = ∫₀^{2π}∫₀^∞ e^{−r²} r dr dθ = 2π·½ = π, so I = √π.
Cylindrical coordinates: formulas and volume element
x = r cosθ, y = r sinθ, z = z, with dV = r dz dr dθ. Best for cylinders, cones, and paraboloids — anything with an axis of symmetry.
Spherical coordinates: formulas and volume element
x = ρ sinφ cosθ, y = ρ sinφ sinθ, z = ρ cosφ, with ρ ≥ 0, φ ∈ [0,π] from the +z axis, θ ∈ [0,2π); dV = ρ² sinφ dρ dφ dθ.
What is the Jacobian's role in a change of variables?
For a transformation from (u,v) to (x,y), dA = |∂(x,y)/∂(u,v)| du dv — the determinant of the matrix of partial derivatives measures local area (or volume) stretching.
