Numerical double integral
Set the polar region and integrand
Use x and y in the Cartesian integrand. Supported functions are sin, cos, tan, asin, acos, atan, sqrt, abs, exp, ln, and log.
Example Data Table
| Cartesian integrand | Polar bounds | Expected value | Meaning |
|---|---|---|---|
1 |
0 ≤ r ≤ 2, 0 ≤ theta ≤ 2π | 4π ≈ 12.5664 | Area of a disk with radius 2. |
x^2 + y^2 |
0 ≤ r ≤ 1, 0 ≤ theta ≤ 2π | π / 2 ≈ 1.5708 | Radial density over a unit disk. |
1 |
1 ≤ r ≤ 3, 0 ≤ theta ≤ π / 2 | 2π ≈ 6.2832 | Area of a quarter annulus. |
Formula Used
The coordinate conversion is x = r cos(theta) and y = r sin(theta). The area element changes from dA to r dr dtheta.
The multiplier r is the Jacobian. It adjusts each narrow polar cell for its changing width.
How to Use This Calculator
- Enter a short label for the circular, annular, or sector-shaped region.
- Type the original density using x and y. Use parentheses for clear grouping.
- Set the smallest and largest radius. The calculator requires nonnegative radii.
- Choose degrees or radians, then enter the beginning and ending angle.
- Select slice counts. Increase them gradually to test numerical stability.
- Press Convert and Calculate. Review the transformed integrand and estimated value.
- Download the CSV or PDF report when you need a saved calculation record.
Why Polar Form Helps
Polar coordinates describe a point by distance and direction. The distance is r. The direction is theta. This view is useful when a region is round, wedge shaped, or centered near the origin. A rectangular description may hide that natural structure. Polar form often makes the boundary rules shorter and easier to inspect.
A double integral measures accumulated quantity over an area. The quantity can represent area, mass, charge, probability, or another density. In Cartesian form, the small area is dx dy. In polar form, the small area changes. It becomes r dr dtheta. The added r is called the Jacobian factor. It prevents small rings near the origin from being treated like wider rings farther away.
Use polar form when boundaries involve circles, radii, or angles. A disk has a constant radial limit. An annulus has an inner and outer radius. A sector uses a start angle and an end angle. These shapes can become simple limits for r and theta. The method can reduce split regions and awkward square roots.
Start by writing the integrand in x and y. Replace x with r cos theta. Replace y with r sin theta. Multiply the complete transformed integrand by r. Then choose the radial and angular limits. Angles must be measured in radians during integration. This calculator converts degree bounds automatically before evaluating the estimate.
The numerical result uses midpoint sampling. The selected region is divided into small radial and angular cells. The calculator evaluates the integrand at each cell midpoint. It multiplies by the cell area. Then it adds every contribution. More slices usually improve the estimate. They also require more processing time. Compare several settings when precision matters.
A numerical estimate is not a symbolic proof. It is still useful for checking algebra, testing limits, and comparing alternate setups. For a constant integrand of one, the result should equal the region area. That provides a helpful verification step. For radial integrands, simplify before entering the expression whenever possible.
Review the displayed converted integrand after calculation. Confirm the Jacobian appears once. Confirm r stays nonnegative. Confirm the angle interval travels through the intended sector. For full circles, use zero to two pi radians, or zero to three hundred sixty degrees. Negative angles are valid when they match the chosen orientation.
Consider discontinuities before relying on any numerical method. A function can fail at isolated points or along a boundary. Some singular expressions need special analysis. Increase the slice count gradually rather than choosing an extreme value immediately. Stable results across repeated runs build confidence. Meaningful changes may reveal unsuitable limits, insufficient sampling, or an expression entered incorrectly during early setup checks.
The exported result records the inputs, converted expression, converted limits, sampling method, and estimate. Keep those details with homework, reports, or design notes. Clear inputs make later checking easier. Polar thinking makes curved regions simpler, clearer, and accurate.
Frequently Asked Questions
What does this calculator convert?
It converts a Cartesian double-integral setup into polar coordinates and estimates the resulting integral with midpoint sampling.
Why is there an extra r factor?
The extra r is the Jacobian. It adjusts the area of each polar cell because wider rings cover more area than rings near the origin.
Can I enter angles in degrees?
Yes. Choose Degrees, then enter the start and end values. The calculator converts them to radians before integration.
Which functions can I use in the integrand?
You can use sin, cos, tan, asin, acos, atan, sqrt, abs, exp, ln, and log. Use x, y, pi, parentheses, and the power symbol ^.
Can the radius begin below zero?
No. This calculator uses the standard polar convention with nonnegative radius bounds. Describe the needed region with appropriate angles instead.
How many slices should I choose?
Start with 80 radial and 160 angular slices. Increase both values and compare estimates when you need stronger numerical confidence.
Does it provide an exact symbolic answer?
No. It provides a numerical midpoint estimate. Use symbolic integration separately when an exact antiderivative or formal proof is required.
How do I calculate a full disk?
Set the radius from zero to the disk radius. Use angles from zero to 360 degrees, or zero to 2π radians.
How do I calculate an annulus?
Enter the inner radius as the minimum radius and the outer radius as the maximum radius. Then choose the full or partial angle range.
What happens with a discontinuous integrand?
The calculation may fail or give an unreliable estimate near singularities. Split the region or use analytical methods when the density is not well behaved.
What do the download buttons include?
The CSV and PDF reports include the input expression, converted polar form, limits, slice counts, and numerical estimate.
Polar thinking makes curved regions simpler, clearer, and accurate.