Double Integrals In Polar Coordinates Calculator

Polar Double Integral Formula:

\[ \iint\limits_{D} f(r,\theta) \, dA = \int_{\theta_1}^{\theta_2} \int_{r_1(\theta)}^{r_2(\theta)} f(r,\theta) \, r \, dr \, d\theta \]

Double Integral Result:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What Are Polar Double Integrals?

Double integrals in polar coordinates are used to integrate over circular or sector-shaped regions. They are particularly useful when dealing with problems that have circular symmetry.

2. How Does the Calculator Work?

The calculator uses the polar double integral formula:

\[ \iint\limits_{D} f(r,\theta) \, dA = \int_{\theta_1}^{\theta_2} \int_{r_1(\theta)}^{r_2(\theta)} f(r,\theta) \, r \, dr \, d\theta \]

Where:

\( f(r,\theta) \) — The function to integrate
\( r \) — Radial distance from origin
\( \theta \) — Angle from polar axis
\( r \, dr \, d\theta \) — Area element in polar coordinates

Explanation: The extra factor of r accounts for the fact that area elements in polar coordinates get larger as you move further from the origin.

3. Importance of Polar Coordinates

Details: Polar coordinates simplify integration over circular regions, making calculations easier for problems with radial symmetry like heat distribution in circular plates or electric fields around point charges.

4. Using the Calculator

Tips:

Enter the function using r and θ as variables (e.g., "r^2 * cos(θ)")
Specify constant or function bounds for r (e.g., "0" to "1" or "0" to "2*cos(θ)")
Enter θ bounds in radians (e.g., "0" to "2*pi" for full circle)

5. Frequently Asked Questions (FAQ)

Q1: Why is there an extra r in polar integrals?
A: The r accounts for the Jacobian determinant in the coordinate transformation, representing how area scales in polar coordinates.

Q2: When should I use polar coordinates?
A: Use polar coordinates when the region of integration is circular or sector-shaped, or when the integrand has radial symmetry.

Q3: Can I integrate any function in polar coordinates?
A: Yes, but it's most beneficial when either the region or the integrand has circular symmetry.

Q4: How do I convert Cartesian to polar coordinates?
A: Use x = r·cos(θ), y = r·sin(θ), and replace dx dy with r dr dθ.

Q5: What are common mistakes in polar integration?
A: Forgetting the r factor, mixing angle units (degrees vs radians), and incorrect bounds are common errors.