Integral of Polar Function Calculator

Calculator Inputs

Choose a polar model, set the interval, then compute integral, area, arc length, and summary metrics in one step.

Function Type

Angle Unit

Decimal Precision

Starting Angle

Ending Angle

Simpson Subintervals

Coefficient a

Coefficient b

Coefficient c

Coefficient d

Coefficient e

Exponent n

Displayed Sample Points

Example Data Table

These examples show how different polar models affect the definite integral, sector area, and arc length across selected intervals.

Function	Interval	Definite Integral	Sector Area	Arc Length
r(θ) = 3 + 2 sin(θ)	0 to π/2	6.712389	14.639380	7.105739
r(θ) = 1 + 0.5θ	0 to π	5.608994	5.330126	5.838850
r(θ) = θ²	0 to 1	0.333333	0.100000	1.060113

Formula Used

Definite integral: I = ∫_θ₁^θ₂ r(θ) dθ

Polar sector area: A = 1/2 ∫_θ₁^θ₂ [r(θ)]² dθ

Average radius: r̄ = I / (θ₂ - θ₁)

RMS radius: r_rms = √[(1 / (θ₂ - θ₁)) ∫_θ₁^θ₂ [r(θ)]² dθ]

Arc length: L = ∫_θ₁^θ₂ √([r(θ)]² + [dr/dθ]²) dθ

This calculator evaluates the chosen polar model numerically with Simpson’s rule. That method is accurate for smooth functions and works well for trigonometric, polynomial, linear, exponential, and mixed expressions.

The integral reports the accumulation of radius over angle. The area reports the swept region, while the arc length follows the actual curve traced by the polar function.

How to Use This Calculator

Select a function family that matches the polar curve you want to study.
Enter the angle unit, starting angle, and ending angle for the interval.
Fill in the coefficients a, b, c, d, e, and exponent n as needed.
Choose the number of Simpson subintervals for numerical accuracy.
Set how many sample points you want displayed in the result table.
Press Calculate Integral to show the results above the form and use the export buttons for CSV or PDF.

FAQs

1. What does this calculator compute?

It calculates the definite integral of r(θ), polar sector area, arc length, average radius, RMS radius, and sampled radius values over your chosen angular interval.

2. Why does it ask for several coefficients?

Different polar models need different parameters. Unused coefficients can stay at zero or default values without affecting models that do not reference them.

3. When should I use degrees instead of radians?

Use degrees when your problem statement is written in degrees. Use radians when formulas, graphing tools, or theoretical work already use radian measure.

4. What is Simpson’s rule doing here?

Simpson’s rule approximates the integral by fitting smooth parabolic segments across many small subintervals. It generally gives strong accuracy for well-behaved polar functions.

5. Why is the ending angle required to be larger?

The calculator uses an ordered interval for clear geometric interpretation. Requiring a larger ending angle avoids confusion in area, arc length, and summary statistics.

6. Can the radius become negative?

Yes. Polar curves can produce negative radius values, which reflect points across the origin. The integral uses the actual signed radius, while the area squares it.

7. How many subintervals should I choose?

Start with 1000 for smooth curves. Increase the count when the curve oscillates rapidly or contains steep changes, then compare whether the results stabilize.

8. What do the CSV and PDF downloads include?

The exports include the selected model, interval, summary metrics, and the sample angle-radius table so you can save, share, or reuse the analysis.