Calculator Inputs
Example Data Table
Use these values to test common rectangular regions.
| Example | Integrand | x Bounds | y Bounds | What to Expect |
|---|---|---|---|---|
| First quadrant strip | x^2 + y^2 | 1 to 3 | 0 to 2 | Positive angles and piecewise radial limits. |
| Centered square | 1 | -2 to 2 | -2 to 2 | Full angle sweep from 0 to 2π. |
| Upper rectangle | x*y | -1 to 2 | 1 to 4 | Angles remain above the x-axis. |
| Left-side region | sqrt(x^2 + y^2) | -4 to -1 | -2 to 2 | Angles appear around π and near wrap points. |
Formula Used
Coordinate change: x = r cos θ and y = r sin θ.
Jacobian: dA = r dr dθ.
Rectangular constraints: x₁ ≤ r cos θ ≤ x₂ and y₁ ≤ r sin θ ≤ y₂.
Polar integral:
∫∫R f(x,y) dA = Σ ∫θ=a to b ∫r=lower(θ) to upper(θ) f(r cos θ, r sin θ) r dr dθ
The calculator splits the rectangle into angular parts when a corner or axis changes the active radial boundary.
How to Use This Calculator
- Enter the integrand using x and y.
- Enter the lower and upper x bounds.
- Enter the lower and upper y bounds.
- Choose numeric samples for estimation accuracy.
- Press the convert button.
- Read the piecewise θ and r limits.
- Download the CSV or PDF report if needed.
Understanding Rectangular to Polar Integral Conversion
Why Convert the Integral?
Rectangular integrals are useful for boxes and grids. Polar integrals are better for radial motion. They also help with circular distance. Many area and mass problems become simpler. The main change is the coordinate system. Every point is written with radius and angle. The radius measures distance from the origin. The angle measures direction from the positive x-axis.
What the Calculator Does
This tool accepts constant rectangular bounds. It reads the interval for x. It also reads the interval for y. Then it tests angular breakpoints. Corners often create new limits. Axes can also change the active boundary. The result may need several polar pieces. That is normal for rectangular regions.
Why the Jacobian Matters
The area element changes during conversion. A small polar sector is not a square. Its size depends on the radius. This creates the extra factor r. Without this factor, the integral is wrong. The calculator shows the transformed integrand. It places r after the new function.
Reading the Output
Each table row is one polar piece. The angle starts at the first value. It ends at the second value. The lower radius begins the ray. The upper radius exits the rectangle. Together, all rows rebuild the same region. The graph helps confirm the shape. The estimates compare both integral forms.
Practical Uses
Students can check homework steps. Teachers can prepare example regions. Engineers can review radial models. Researchers can compare numerical estimates. The export buttons save clean results. The examples give quick test cases. Use more samples for smoother estimates. Use fewer samples for faster checks.
FAQs
1. What does this calculator convert?
It converts a double integral over constant rectangular bounds into polar form. It also estimates the integral in both coordinate systems.
2. Why does the answer have many pieces?
A rectangle usually does not match one simple polar sector. Different angles may hit different sides, so the radius limits change.
3. What is the Jacobian factor?
The Jacobian factor is r. It appears because polar area elements grow as the radius grows.
4. Can I enter trigonometric functions?
Yes. You can enter functions like sin(x), cos(y), sqrt(x^2+y^2), log(x), exp(y), and similar supported expressions.
5. Are the numeric estimates exact?
No. They are midpoint numerical estimates. Increase the sample count for better accuracy, but very high values may run slower.
6. What bounds should I enter?
Enter constant bounds only. Use x minimum, x maximum, y minimum, and y maximum to define the rectangular region.
7. Why do degrees and radians both appear?
Radians are standard for calculus. Degrees help you visually understand the angular span of each polar piece.
8. What do the download buttons save?
The CSV saves the piecewise polar bounds. The PDF saves a readable summary with inputs, transformed integrand, and estimates.