Exploring the Area of the Cardioid ( r 2 2sintheta ) in Polar Coordinates

In this article, we will delve into how to find the area enclosed by a cardioid given in polar coordinates by the equation ( r 2 2sintheta ). We'll walk through the steps from setting up the integral to calculating the final area, providing a comprehensive guide for anyone dealing with similar polar equations.

Introduction to Polar Coordinates

Polar coordinates offer a unique way to describe points in the plane, using a distance ( r ) from a fixed origin (the pole) and an angle ( theta ) from a fixed line. When we need to find the area enclosed by a curve defined in polar coordinates, we use the integral formula:

[ A frac{1}{2} int_{alpha}^{beta} r^2 , dtheta ]

Step-by-Step Solution

Step 1: Determine the Limits of Integration

The cardioid ( r 2 2sintheta ) is symmetric about the line ( theta frac{pi}{2} ). Since the curve completes one full rotation from ( theta 0 ) to ( theta pi ), we will integrate over this interval.

Step 2: Set Up the Integral

With the limits ( alpha 0 ) and ( beta pi ), our integral for the area is:

[ A frac{1}{2} int_{0}^{pi} (2 2sintheta)^2 , dtheta ]

Step 3: Simplify the Expression for ( r^2 )

First, expand the expression ( (2 2sintheta)^2 ):

[ (2 2sintheta)^2 4 8sintheta 4sin^2theta ]

Step 4: Substitute and Simplify the Integral

Substitute the simplified expression into the area integral:

[ A frac{1}{2} int_{0}^{pi} left(4 8sintheta 4sin^2thetaright) , dtheta ]

This can be split into three separate integrals:

[ A frac{1}{2} left[ int_{0}^{pi} 4 , dtheta int_{0}^{pi} 8sintheta , dtheta int_{0}^{pi} 4sin^2theta , dtheta right] ]

Step 5: Calculate Each Integral

First Integral

[ int_{0}^{pi} 4 , dtheta 4theta Bigg|_{0}^{pi} 4pi ]

Second Integral

[ int_{0}^{pi} 8sintheta , dtheta -8costheta Bigg|_{0}^{pi} -8(-1 - 1) 16 ]

Third Integral

Using the identity ( sin^2theta frac{1 - cos2theta}{2} ):

[ int_{0}^{pi} 4sin^2theta , dtheta 4 cdot frac{1}{2} int_{0}^{pi} left(1 - cos2thetaright) , dtheta ]

[ 2 left[theta - frac{1}{2} sin2thetaright] Bigg|_{0}^{pi} 2left(pi - 0right) 2pi ]

Step 6: Combine the Results

Now, add up the results of the integrals:

[ A frac{1}{2} left(4pi 16 2piright) frac{1}{2} left(6pi 16right) 3pi 8 ]

Final Answer

The area enclosed by the cardioid ( r 2 2sintheta ) is boxed{3pi 8}.

Extension and Further Exploration

Since this is a polar curve, we can explore the area in a different manner by symmetrically considering the region from ( -frac{pi}{2} ) to ( frac{pi}{2} ). The integral in this case would be:

[ A 2 int_{-frac{pi}{2}}^{frac{pi}{2}} frac{1}{2} int_{0}^{2 2sintheta} r , dr , dtheta ]

This method would yield the same result, providing an additional approach to solving the problem and reinforcing the accuracy of our previous solution.

Conclusion

Understanding how to find the area of a cardioid in polar coordinates is a valuable skill in mathematical analysis. By breaking down the integral into manageable parts and carefully evaluating each section, we can derive the total area enclosed by the curve. This method can be applied to a wide range of polar equations, offering a robust toolset for various mathematical applications.