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Circular symmetry

Polar and cylindrical coordinates

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

x=rcos⁡θ, y=rsin⁡θ,dA=r dr dθx=r\cos\theta,\ y=r\sin\theta,\quad dA=r\,dr\,d\theta

Polar coordinates simplify disks, sectors and annuli. Use r ≥ 0 and an angular interval that does not count the region more than once. Cylindrical coordinates retain z and the volume element is r dr dθ dz. Enter the original integrand in x, y, z: the calculator performs the substitution and adds the Jacobian.

Common mistake

Forgetting the r factor or manually adding it when the calculator already includes it.

DEVELOPED GUIDE · 8 min

Recognize when polar coordinates simplify a region and add the Jacobian exactly once.

When to choose this method

  • The region contains disks, sectors or annuli centered at the origin.
  • The expression contains x²+y² or rotational symmetry. An offset disk may require dependent bounds.

Before calculating

  1. Transform the function and describe the radius and angle.
  2. Multiply by r to turn dA into r dr dθ.

A complete example, step by step

∬x2+y2≤4(x2+y2) dA\iint_{x^2+y^2\le4}(x^2+y^2)\,dA

  1. The region is the full disk of radius 2: all angles are needed.

    0≤r≤2,0≤θ≤2π0\le r\le2,\qquad0\le\theta\le2\pi

  2. Substitute x=r cosθ and y=r sinθ into the function.

    x2+y2=r2(cos⁡2θ+sin⁡2θ)=r2x^2+y^2=r^2(\cos^2\theta+\sin^2\theta)=r^2

  3. The function and Jacobian are separate: here the function is r² and the Jacobian supplies another r.

    I=∫02π∫02r3 dr dθI=\int_0^{2\pi}\int_0^2 r^3\,dr\,d\theta

  4. Integrate with respect to radius and apply its bounds.

    [r44]02=4\left[\frac{r^4}{4}\right]_0^2=4

  5. The angular integral multiplies by the full turn.

    I=∫02π4 dθ=8πI=\int_0^{2\pi}4\,d\theta=8\pi

Check the result and domain

The function is continuous and nonnegative on a compact disk. Enter x²+y² as the original function in the calculator; the engine adds r automatically.

dA=r dr dθ,14π∬D(x2+y2) dA=2dA=r\,dr\,d\theta,\qquad\frac1{4\pi}\iint_D(x^2+y^2)\,dA=2
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Now try it yourself

Solve on paper before opening the hint or answer. These are self-assessment activities; to check a typed answer, open Learn.

In these exercises θ runs from 0 to 2π: a full revolution is intended.

EXERCISE 1∬x2+y2≤91 dA\iint_{x^2+y^2\le9}1\,dA
Show a hint

This is the area of a radius-3 disk.

Check my result9π9\pi
EXERCISE 2∬x2+y2≤1(x2+y2) dA\iint_{x^2+y^2\le1}(x^2+y^2)\,dA
Show a hint

Integrate r³ from 0 to 1.

Check my resultπ2\frac\pi2
EXERCISE 3∬1≤x2+y2≤41 dA\iint_{1\le x^2+y^2\le4}1\,dA
Show a hint

The annulus has 1≤r≤2.

Check my result3π3\pi
Continue with interactive practice

Connect with other methods

General variable changes and the Jacobian →Cylindrical coordinates →Spherical coordinates →

Keep learning

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