Problem 56 - Essential Calculus

Question

Find the area of the circle given by \(r=\sin \theta+\cos \theta\). Check your result by converting the polar equation to rectangular form, then using the formula for the area of a circle.

Step-by-Step Solution

Answer

The area of the given circle is \(2\pi\) square units

Step 1: Convert the Polar Equation to the Rectangular Coordinate System

The conversion from polar coordinates to rectangular coordinates is done using the relations \(x = r\cos\theta\) and \(y = r\sin\theta\). Substituting \(r = \sin\theta+\cos\theta\) into these relations, we get:\[x=(\sin \theta +\cos \theta)cos\theta = \sin \theta \cos\theta+\cos^2 \theta\]\[y=(\sin \theta +\cos \theta)sin\theta = \sin^2 \theta + \cos \theta\sin\theta \]Squaring and adding these two equations,we get:\[ x^2 + y^2 = (\sin^2 \theta + 2\cos \theta\sin\theta + \cos^2 \theta) = 1 + \sin 2\theta \]

Step 2: Determine the Radius

Observe that the derived equation corresponds to the square of a typical circle equation in rectangular coordinates \( x^2 + y^2 = r^2 \). From this, we find that the radius is given by \( r = \sqrt{1+\sin 2\theta}\). However, the maximum value of \(\sin 2\theta\) is 1, so the maximum radius is \( r = \sqrt{1+1}= \sqrt{2}\).

Step 3: Computing the Area

The standard formula for the area of a circle is \(A = \pi r^2\). By substituting \(r = \sqrt{2}\) (the maximum value) into this formula, we get:\[A = \pi (\sqrt{2})^2 = 2\pi\]

Key Concepts

Circle Area Calculation

Calculating the area of a circle is a fundamental skill in geometry. The area of a circle can be found using the simple formula:

• \( A = \pi r^2 \)

where \( r \) represents the radius of the circle. This formula arises from the geometric principle that the area of a circle is proportional to the square of its radius, with the constant of proportionality being \( \pi \).

When dealing with more complex forms of a circle, such as those represented in polar coordinates, you may need to first find the radius before using the area formula. In our exercise, after converting the polar equation to rectangular form, we determined that the maximum radius \( r = \sqrt{2} \) for the given circle. Using the formula, the area was calculated as:

• \( A = \pi (\sqrt{2})^2 = 2\pi \)

This confirms the consistency of the circle's area, regardless if it's viewed in polar or rectangular coordinates.

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a radius and an angle. Instead of the standard x-y grid, polar coordinates use:

• \( r \): the distance from the origin (center point)
• \( \theta \): the angle from the positive x-axis

The benefit of using polar coordinates is their ease in describing circular and rotational motion, where angles naturally play a significant role.

In the given exercise, the equation \( r = \sin \theta + \cos \theta \) is expressed in polar form. The task was to translate this form into its rectangular coordinate equivalent, facilitating the usage of other geometric relationships in problem-solving. Understanding the basis of polar coordinates is essential for converting between these coordinate systems effectively.

Rectangular Coordinates

Rectangular coordinates, also called Cartesian coordinates, describe a point in a plane by its distance along the horizontal (x) and vertical (y) axes. They are the most common system used for graphing in algebra and geometry.

• To convert polar to rectangular coordinates, use the formulas \( x = r \cos \theta \) and \( y = r \sin \theta \).

This system makes it easy to interpret positions and shapes, especially for linear equations and systems.

In our exercise, we began with the polar equation \( r = \sin \theta + \cos \theta \) and used the conversion formulas to find the corresponding rectangular form. By doing so, we identified the circle's equation as \( x^2 + y^2 = 1 + \sin 2\theta \). This allowed us to determine the circle's radius geometrically, and hence compute its area with familiar methods. Mastery of moving between polar and rectangular coordinates enhances versatility in solving a variety of geometrical problems.