For a region described in polar bounds, a standard setup is:
∬R f(x,y) dA = ∫θ=ab ∫r=g1(θ)g2(θ) f(r cosθ, r sinθ) · r dr dθ
The extra r is the Jacobian for converting area from Cartesian to polar coordinates. You can toggle it off to test understanding, but keep it on for real polar integrals.
- Type your integrand using
r,theta, orx,y. - Enter
rbounds as constants or expressions intheta. - Set
θstart/end (usepiif needed). - Choose degrees only if your θ inputs are in degrees.
- Increase subdivisions for tougher integrals or sharper boundaries.
- Press Calculate; results appear above the form under the header.
- Download CSV/PDF to save the computed summary and samples.
Example: area of a disk with radius 2 (f=1, Jacobian included).
| Input | Value | Meaning |
|---|---|---|
| f(r,θ) | 1 | Area integral (integrand equals 1) |
| r lower | 0 | From the origin |
| r upper | 2 | To radius 2 |
| θ start | 0 | Full rotation begins |
| θ end | 2*pi | Full rotation ends |
| Expected value | 4*pi ≈ 12.5663706144 | Area of a circle, πr² with r=2 |
Double Integral Polar Coordinates Guide
1) Why polar double integrals matter
Many circular regions are awkward in Cartesian form but simple in polar form. Disks, annuli, sectors, and regions bounded by curves like r=2*cos(theta) become clean limits. This calculator helps you evaluate those setups numerically when a closed-form antiderivative is difficult.
2) What this calculator computes
The tool evaluates a polar double integral by nesting two numerical integrals: an inner integral in r and an outer integral in theta. You enter f(r,theta) (or use x,y), plus limits for r and theta. Results are shown with a refinement check and a small sample table.
3) Understanding the Jacobian factor r
When converting dA from Cartesian to polar, the area element becomes r dr dtheta. That extra r is the Jacobian. If you forget it, areas and mass integrals will be wrong by a scale that grows with radius. Keep the Jacobian switch on for standard polar integrals.
4) Choosing bounds and region shape
Bounds can be constants or functions of theta. For a full disk of radius 2, use r: 0 to 2 and theta: 0 to 2*pi. For an annulus from 1 to 3, use r: 1 to 3. For a right-facing circle, you can set r: 0 to 2*cos(theta) with theta: -pi/2 to pi/2.
5) Accuracy controls and error estimate
The calculator uses composite Simpson’s rule, which requires even subdivisions. The inputs n_r and n_theta control resolution in each direction. A refined run doubles these counts (capped for speed), and the absolute difference |refined - base| is shown as a quick error estimate.
6) Practical examples with expected values
For f=1 on a radius-2 disk, the expected value is area 4*pi ≈ 12.5663706144. For an annulus with r: 1 to 3, the expected area is pi(9-1)=8*pi ≈ 25.1327412287. These checks are useful to verify bounds, Jacobian, and angle units before solving harder integrands.
7) Tips for stable input expressions
Prefer radians unless your problem statement is in degrees, then select degrees for automatic conversion. Use pi, sin(), cos(), sqrt(), and powers like r^2. If you see large error estimates, increase subdivisions gradually and simplify sharp boundaries into correct theta-dependent limits.
FAQs
1) Can r-limits depend on theta?
Yes. Enter expressions like 2*cos(theta) for the upper limit and the calculator evaluates the inner integral using that bound for each theta step.
2) Should I keep the Jacobian switch on?
For standard polar conversion, yes. The correct element is r dr dtheta. Turning it off is only for learning or testing an already adjusted integrand.
3) What if I enter theta end smaller than theta start?
The calculator swaps the bounds automatically and shows a “bounds swapped” note, so you still get a positive orientation for integration.
4) Which functions and variables can I use?
You may use r, theta, x, y, pi, and common math functions like sin(), cos(), sqrt(), log(), and exp().
5) Why do base and refined results differ?
They use different subdivision counts. The difference is a quick accuracy signal. Increase n_r and n_theta if the estimate is large or the function varies rapidly.
6) When should I use degrees mode?
Use degrees mode only when your theta limits are given in degrees. The tool converts them to radians internally so trigonometric functions and integration steps remain consistent.