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06 Multiple Integrals in Curvilinear Coordinates

A practical guide to transforming double and triple integrals with Jacobians in polar, cylindrical, and spherical coordinates, including setup strategies and applications to area, volume, mass, and moments of inertia.

Why coordinate changes simplify multiple integrals

A coordinate system is useful when its variables follow the geometry of the region. Circular and radial planar regions suggest ; solids with symmetry about the zz-axis suggest ; spheres, balls, and cones often suggest .

For a planar transformation

x=x(u,v),y=y(u,v),x=x(u,v),\qquad y=y(u,v),

the area integral becomes

∬Rf(x,y) dA=∬Sf(x(u,v),y(u,v))∣∂(x,y)∂(u,v)∣ du dv.\iint_R f(x,y)\,dA = \iint_S f\bigl(x(u,v),y(u,v)\bigr) \left|\frac{\partial(x,y)}{\partial(u,v)}\right|\,du\,dv.

The three essential tasks are:

  1. Rewrite the integrand in the new variables.

  2. Describe the region and determine bounds in the new variables.

  3. Replace the differential element using the absolute Jacobian.

The same pattern applies to triple integrals, with a three-dimensional Jacobian multiplying the transformed integrand.

Takeaway: Choose coordinates that simplify both the boundaries and the integrand, then transform every part of the integral consistently.

for planar regions

use

x=rcos⁡θ,y=rsin⁡θ,dA=r dr dθ.x=r\cos\theta, \qquad y=r\sin\theta, \qquad dA=r\,dr\,d\theta.

The inverse relation r=x2+y2r=\sqrt{x^2+y^2} makes expressions involving x2+y2x^2+y^2 especially simple because

x2+y2=r2.x^2+y^2=r^2.

Common boundary conversions include:

  • The circle x2+y2=a2x^2+y^2=a^2 becomes r=ar=a.

  • The disk x2+y2≤a2x^2+y^2\le a^2 becomes 0≤r≤a0\le r\le a.

  • A line through the origin, such as y=mxy=mx, becomes a constant angle.

  • The region x2+y2≤2axx^2+y^2\le 2ax becomes 0≤r≤2acos⁡θ0\le r\le 2a\cos\theta.

  • The region x2+y2≤2ayx^2+y^2\le 2ay becomes 0≤r≤2asin⁡θ0\le r\le 2a\sin\theta.

For the disk x2+y2≤a2x^2+y^2\le a^2, the area is

A=∫02π∫0ar dr dθ=πa2.A=\int_0^{2\pi}\int_0^a r\,dr\,d\theta=\pi a^2.

For the disk DD given by x2+y2≤4x^2+y^2\le 4,

∬D(x2+y2) dA=∫02π∫02r2(r dr dθ)=∫02π∫02r3 dr dθ=8π.\begin{aligned} \iint_D (x^2+y^2)\,dA &=\int_0^{2\pi}\int_0^2 r^2(r\,dr\,d\theta)\\ &=\int_0^{2\pi}\int_0^2 r^3\,dr\,d\theta =8\pi. \end{aligned}

Takeaway: In , convert radial expressions to powers of rr, identify the angular sweep, and never omit the factor rr in dAdA.

for solids

are defined by

x=rcos⁡θ,y=rsin⁡θ,z=z,x=r\cos\theta, \qquad y=r\sin\theta, \qquad z=z,

with volume element

dV=r dr dθ dz.dV=r\,dr\,d\theta\,dz.

They are effective for vertical cylinders and solids whose cross-sections are circular. For example,

x2+y2=a2⟶r=a,x^2+y^2=a^2\longrightarrow r=a,

and

x2+y2=z2⟶r2=z2.x^2+y^2=z^2\longrightarrow r^2=z^2.

For a cylinder with x2+y2≤a2x^2+y^2\le a^2 and 0≤z≤h0\le z\le h, use

0≤r≤a,0≤θ≤2π,0≤z≤h.0\le r\le a, \qquad 0\le\theta\le 2\pi, \qquad 0\le z\le h.

Its volume is

V=∫02π∫0a∫0hr dz dr dθ=πa2h.V=\int_0^{2\pi}\int_0^a\int_0^h r\,dz\,dr\,d\theta=\pi a^2h.

For the upper hemisphere x2+y2+z2≤R2x^2+y^2+z^2\le R^2, z≥0z\ge 0, the bounds are

0≤r≤R,0≤z≤R2−r2,0≤θ≤2π.0\le r\le R, \qquad 0\le z\le\sqrt{R^2-r^2}, \qquad 0\le\theta\le 2\pi.

Thus,

V=∫02π∫0R∫0R2−r2r dz dr dθ=23πR3.V=\int_0^{2\pi}\int_0^R\int_0^{\sqrt{R^2-r^2}}r\,dz\,dr\,d\theta =\frac{2}{3}\pi R^3.

Takeaway: separate horizontal circular geometry from vertical bounds, while the factor rr accounts for horizontal area scaling.

for radial and conical regions

use the standard calculus convention

x=ρsin⁡ϕcos⁡θ,y=ρsin⁡ϕsin⁡θ,z=ρcos⁡ϕ,x=\rho\sin\phi\cos\theta, \qquad y=\rho\sin\phi\sin\theta, \qquad z=\rho\cos\phi,

where ρ\rho is the distance from the origin, θ\theta is the azimuthal angle in the xyxy-plane, and ϕ\phi is measured downward from the positive zz-axis. The volume element is

dV=ρ2sin⁡ϕ dρ dϕ dθ.dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta.

Useful conversions are

x2+y2+z2=ρ2,x2+y2=ρ2sin⁡2ϕ,z=ρcos⁡ϕ.x^2+y^2+z^2=\rho^2, \qquad x^2+y^2=\rho^2\sin^2\phi, \qquad z=\rho\cos\phi.

Common bounds include:

  • A sphere x2+y2+z2=a2x^2+y^2+z^2=a^2 becomes ρ=a\rho=a.

  • A ball x2+y2+z2≤a2x^2+y^2+z^2\le a^2 becomes 0≤ρ≤a0\le\rho\le a.

  • The upper half-space z≥0z\ge 0 becomes 0≤ϕ≤π20\le\phi\le\frac{\pi}{2}.

  • The cone z=x2+y2z=\sqrt{x^2+y^2} becomes ϕ=π4\phi=\frac{\pi}{4}.

  • The positive octant has 0≤θ≤π20\le\theta\le\frac{\pi}{2} and 0≤ϕ≤π20\le\phi\le\frac{\pi}{2}.

For the ball x2+y2+z2≤R2x^2+y^2+z^2\le R^2,

V=∫02π∫0π∫0Rρ2sin⁡ϕ dρ dϕ dθ=43πR3.V=\int_0^{2\pi}\int_0^\pi\int_0^R \rho^2\sin\phi\,d\rho\,d\phi\,d\theta =\frac{4}{3}\pi R^3.

For the region inside ρ≤R\rho\le R and above the cone z=x2+y2z=\sqrt{x^2+y^2}, use 0≤ϕ≤π40\le\phi\le\frac{\pi}{4}. Its volume is

V=∫02π∫0π/4∫0Rρ2sin⁡ϕ dρ dϕ dθ=2π(1−22)R33.V=\int_0^{2\pi}\int_0^{\pi/4}\int_0^R \rho^2\sin\phi\,d\rho\,d\phi\,d\theta =2\pi\left(1-\frac{\sqrt{2}}{2}\right)\frac{R^3}{3}.

Takeaway: make radial boundaries and conical angular boundaries constant, but the factor ρ2sin⁡ϕ\rho^2\sin\phi must always be included.

Applications and a setup checklist

Coordinate transformations support geometric and physical applications. Set the integrand equal to 11 for area or volume:

Area⁡(R)=∬R1 dA,Vol⁡(E)=∭E1 dV.\operatorname{Area}(R)=\iint_R1\,dA, \qquad \operatorname{Vol}(E)=\iiint_E1\,dV.

For a lamina with surface density δ(x,y)\delta(x,y), mass is

M=∬Rδ(x,y) dA.M=\iint_R\delta(x,y)\,dA.

For a solid with density δ(x,y,z)\delta(x,y,z), mass is

M=∭Eδ(x,y,z) dV.M=\iiint_E\delta(x,y,z)\,dV.

The of a planar lamina is determined by

xˉ=1M∬Rxδ(x,y) dA,yˉ=1M∬Ryδ(x,y) dA.\bar{x}=\frac{1}{M}\iint_R x\delta(x,y)\,dA, \qquad \bar{y}=\frac{1}{M}\iint_R y\delta(x,y)\,dA.

For a solid, the coordinates are

xˉ=1M∭Exδ dV,yˉ=1M∭Eyδ dV,zˉ=1M∭Ezδ dV.\bar{x}=\frac{1}{M}\iiint_E x\delta\,dV, \quad \bar{y}=\frac{1}{M}\iiint_E y\delta\,dV, \quad \bar{z}=\frac{1}{M}\iiint_E z\delta\,dV.

The about the zz-axis is

Iz=∭E(x2+y2)δ(x,y,z) dV.I_z=\iiint_E(x^2+y^2)\delta(x,y,z)\,dV.

In , this becomes an integral containing r2r^2 from the distance to the axis and another rr from the volume element.

A reliable setup checklist is:

  1. Identify the geometry and choose the coordinate system.

  2. Write the coordinate conversion formulas.

  3. Transform the integrand and every boundary.

  4. Determine angular bounds before radial or vertical bounds.

  5. Insert the correct Jacobian factor.

  6. Check that the region is covered exactly once.

  7. Use symmetry when it reduces the computation.

Takeaway: The most common errors are omitted Jacobian factors, incorrect angular ranges, and bounds that do not describe the intended region.