Area Between Two Polar Curves Double Integral Calculator

How to Use This Calculator

To calculate the area between two polar curves:

1. Enter the first polar curve equation in the "First Curve" field. For example, r = 2 + cos(θ).
2. Enter the second polar curve equation in the "Second Curve" field. For example, r = 1 + sin(θ).
3. Specify the lower and upper bounds for θ (theta) in radians.
4. Click "Calculate" to compute the area.

The calculator will display the area between the two curves in square units.

The Formula

The area between two polar curves r₁(θ) and r₂(θ) from θ = a to θ = b is given by:

Where:

• r₂(θ) is the outer curve
• r₁(θ) is the inner curve
• a and b are the lower and upper bounds for θ

This formula accounts for the fact that polar coordinates involve both radial and angular components, requiring integration over the angular range.

Worked Example

Let's calculate the area between the curves r₁ = 1 and r₂ = 2cos(θ) from θ = 0 to θ = π/2.

1. First, identify the curves and bounds:
   • r₁(θ) = 1
   • r₂(θ) = 2cos(θ)
   • a = 0, b = π/2
2. Set up the integral:
   A = (1/2) ∫[from 0 to π/2] [ (2cos(θ))² - (1)² ] dθ
3. Simplify the integrand:
   A = (1/2) ∫[from 0 to π/2] [4cos²(θ) - 1] dθ
4. Use the identity cos²(θ) = (1 + cos(2θ))/2:
   A = (1/2) ∫[from 0 to π/2] [4(1 + cos(2θ))/2 - 1] dθ
5. Simplify further:
   A = (1/2) ∫[from 0 to π/2] [2 + 2cos(2θ) - 1] dθ = (1/2) ∫[from 0 to π/2] [1 + 2cos(2θ)] dθ
6. Integrate term by term:
   A = (1/2) [θ + sin(2θ)] evaluated from 0 to π/2
7. Calculate the definite integral:
   A = (1/2) [(π/2 + sin(π)) - (0 + sin(0))] = (1/2) [π/2] = π/4

The area between the curves is π/4 square units.