To find the area under a polar curve, you can follow these simple steps:

• Identify the Polar Curve:
  First, look at the polar function ( r = f(\theta) ). This is the equation that shows the curve you want to calculate the area for.

• Set the Limits of Integration:
  Next, choose the angles ( \theta_1 ) and ( \theta_2 ). These angles mark the start and end of the area you're interested in. They should be the points where the curve crosses itself or the center point (called the pole).

• Use the Area Formula:
  To find the area ( A ) inside the polar curve between ( \theta_1 ) and ( \theta_2 ), use this formula:

  $$A = \frac{1}{2} \int_{\theta_1}^{\theta_2} (f(\theta))^2 \, d\theta$$

  This means we’re adding up tiny slices of the curve to get the total area.

• Evaluate the Integral:
  Now, calculate the integral of ( (f(\theta))^2 ) between the angles you set earlier. This will give you the total area under the curve.

• Consider Multiple Loops (if needed):
  If your polar curve has more than one loop, break the area into parts. Find the area for each part using the same steps, and then add them together.

• Account for Symmetry:
  If the curve is symmetrical (looks the same on both sides), you can use this to make your calculations easier. For example, if the curve is symmetrical around the x-axis, you can find the area from ( 0 ) to ( \theta = \frac{\pi}{2} ) and then multiply that area by 4 to get the total.

• Check Units and Context:
  Finally, make sure your area is reported in the right units based on your problem. Areas found using polar coordinates usually are in square units.

By following these steps, you can easily find the area under any polar curve. This technique is useful in calculus and other related subjects.