Problem 12 - The Calculus with Analytic Geometry

Question

Find the area of the intersection of the regions enclosed by the graphs of the two given equations.\(\left\\{\begin{array}{l}r=4 \sin \theta \\ r=4 \cos \theta\end{array}\right.\)

Step-by-Step Solution

Answer

The area is \2\pi - 4.

Step 1: Convert Equations to Cartesian Form

Convert the polar equations to their Cartesian equivalents. For the first equation, \(r = 4 \sin(\theta)\), use the Cartesian form \(r = \sqrt{x^2 + y^2}\), \(x = r \cos(\theta)\), and \(y = r \sin(\theta)\). This results in the equation \(r = 4 \sin(\theta) \Rightarrow \sqrt{x^2 + y^2} = 4 \frac{y}{\sqrt{x^2 + y^2}} \Rightarrow x^2 + y^2 = 4y\).

Step 2: Simplify Second Equation

For the second equation, \(r = 4 \cos(\theta)\), use the same approach. \(r = 4 \cos(\theta) \Rightarrow \sqrt{x^2 + y^2} = 4 \frac{x}{√{x^2 + y^2}} \Rightarrow x^2 + y^2 = 4x\).

Step 3: Simplify to Standard Form

Simplify the Cartesian equations further to obtain the forms: \(x^2 + y^2 - 4y = 0\) and \(x^2 + y^2 - 4x = 0\). These can be rewritten as: \( (x - 0)^2 + (y - 2)^2 = 4\) and \( (x - 2)^2 + (y - 0)^2 = 4\). These are circles centered at (0, 2) and (2, 0) with radius 2.

Step 4: Find Points of Intersection

Solve the system of equations \(x^2 + y^2 = 4y\) and \(x^2 + y^2 = 4x\) to find points of intersection. Subtracting one from the other, \(4x = 4y\), or \(x = y\). Substitute \(x = y\) into one of the original equations: \(x^2 + x^2 = 4x\Rightarrow 2x^2 = 4x\Rightarrow 2x(x-2) = 0\Rightarrow x = 0\) or \(x = 2\). Thus, the intersections are at points (0, 0) and (2, 2).

Step 5: Set Up the Integral

To find the area of intersection, set up the integral in polar coordinates where the integration bounds are from \(\theta = 0\) to \(\theta = \frac{\pi}{4}\). The area A is: \(A = 2 \times \int_{0}^{\frac{\pi}{4}} \frac{1}{2} (4 \sin (\theta))^2 d\theta\).

Step 6: Evaluate the Integral

Simplify and evaluate the integral: \(A = 2 \times \int_{0}^{\frac{\pi}{4}} 8 \sin^2 (\theta)d\theta = 16 \int_{0}^{\frac{\pi}{4}} \frac{1 - \cos (2\theta)}{2} d\theta\). This splits into two simpler integrals: \(8 \int_{0}^{\frac{\pi}{4}} 1 d\theta - 8 \int_{0}^{\frac{\pi}{4}} \cos (2\theta) d\theta\).

Step 7: Final Integration and Simplification

Evaluate each part: \(8 \left[ \theta \right]_0^{\frac{\pi}{4}} - 8 \left[ \frac{1}{2} \sin (2\theta) \right]_0^{\frac{\pi}{4}} = 8 \left( \frac{\pi}{4} - 0 \right) - 4 \left( \sin \left(\frac{\pi}{2}\right) - \sin (0) \right) = 2\pi - 4\).

Step 8: Simplify Result

Simplify the final result: \(2(\frac{\pi}{2} - 1) = \2\pi - 4\).

Key Concepts

Polar Coordinates

Polar coordinates offer a way to describe points on a plane using angles and distance from a central point.
Unlike Cartesian coordinates which use \(x\) and \(y\) values, polar coordinates use \(r\) and \(\theta\).
Here, \(r\) is the distance from the origin to the point, and \(\theta\) is the angle from the positive x-axis to the line that connects the origin to the point.
This is particularly useful for dealing with equations involving circles and periodic functions.

In this exercise, the given polar equations represent circles when converted to Cartesian form.

Cartesian Coordinates

Cartesian coordinates (\(x, y\)) are the most familiar way to represent points on a plane.
Here, each point is described by its horizontal distance \(x\) and vertical distance \(y\) from the origin.
Converting from polar to Cartesian coordinates involves relationships such as:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)
• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \arctan(\frac{y}{x})\)

This conversion simplifies finding the area of overlapping regions, as shown in the provided exercise.

Integral Calculus

Integral calculus involves finding areas under curves, among other applications.
In this problem, the area of the region where two circles intersect is found using double integrals in polar coordinates.
The double integral accounts for the changing radius depending on the angle \(\theta\).

The integral set-up for the problem is:

• \(A = 2 \times \int_{0}^{\frac{\pi}{4}} \frac{1}{2} \left(4 \sin(\theta)\right)^2 \, d\theta\)

Simplifying and evaluating this integral yields the desired intersection area.

Geometric Intersection

Geometric intersection determines where two shapes overlap.
In this exercise, we find where two circles intersect. This includes solving for the points where their equations are equal.
Converting both polar equations, \(r = 4 \sin(\theta)\) and \(r = 4 \cos(\theta)\), to Cartesian form results in circles:

• \(x^2 + y^2 = 4y\)
• \(x^2 + y^2 = 4x\)

Solving these simultaneously gives intersection points at (0,0) and (2,2).
This sets the bounds for the integral to find the intersection area.

Circle Equations

Circle equations are critical in this problem.
A circle's standard form is \((x - h)^2 + (y - k)^2 = r^2\), where (h,k) is the center and r is the radius.
The given polar equations convert to:

• \((x-0)^2 + (y-2)^2 = 4\)
• \((x-2)^2 + (y-0)^2 = 4\)

Indicating two circles centered at (0,2) and (2,0) with a radius of 2.
Understanding circle equations helps in identifying intersecting areas and setting up integrals for calculation.