Problem 26 - Essential Calculus

Question

Use a graphing utility to graph the polar equations and find the area of the given region. Inside \(r=3 \sin \theta\) and outside \(r=2-\sin \theta\)

Step-by-Step Solution

Answer

After computing the definite integral described in step 4, the area of the given region, correct to two decimal places, is 4.50 square units.

Step 1: Graph the polar functions

The first step involves graphing the polar equations \(r = 3\sin\theta\) and \(r = 2 - \sin\theta\). By using a graphing tool, the plots of these functions can be visualized. It is clear that the two graphs intersect

Step 2: Find the points of intersection

The next step is to find the exact points of where these plots intersect. For this, first equate \(3\sin\theta\) and \(2 - \sin\theta\) and solve for \(\theta\). This gives us \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\) as the points where the two curves intersect

Step 3: Determine the limits of integration

The points of intersection serve as the limits of integration. As per the regions defined by the boundaries, the integration will be from \(\frac{\pi}{2}\) to \(\frac{3\pi}{2}\)

Step 4: Compute the area

Now, you must compute the area using the formula for polar area. As we have to find the area inside \(r = 3\sin\theta\) and outside \(r = 2 - \sin\theta\), the corresponding formula for the area will be \(A = \frac{1}{2}\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} [(3\sin\theta)^2 - (2 - \sin\theta)^2] d\theta \). Solving this using integral calculus gives a numerical value for the area

Key Concepts

Polar Graphs

Polar graphs represent equations or curves using polar coordinates.Instead of using traditional Cartesian coordinates (x, y), polar graphs rely on coordinates defined by a radius and an angle.The polar coordinate system consists of concentric circles and rays starting from a central point known as the pole.
The angle, known as \(\theta\), is measured in radians from the positive x-axis, while the radius \(r\) tells the distance from the pole.For example, the equation \(r = 3\sin\theta\) generates a graph of a circle displaced above the pole, while \(r = 2-\sin\theta\) represents a limaçon.

• Visualizing polar graphs can help us understand their interaction and overlap.
• Graphing utilities like software or calculators offer easy plotting of these curves.

Area Calculation in Polar Coordinates

Calculating areas in polar coordinates involves understanding the intricacies of the curves.The formula for area in polar coordinates is quite different from Cartesian coordinates.
To calculate the area enclosed between two polar graphs, you integrate within the bounds defined by their intersections.This involves determining the limits of integration, which are usually the intersection points along \(\theta\).

• These bounds help partition the region you wish to calculate between two curves.
• The overlapping region's shape affects the integration boundaries and complexity of computation.

For the problem at hand, we find the area inside one curve and outside the other by using the formula:\[ A = \frac{1}{2} \int_{\text{start}}^{\text{end}} \left[ f(\theta)^2 - g(\theta)^2 \right] d\theta \]This expression captures the net area difference between the squared functions, offering insight into their spatial overlap.

Integration in Polar Coordinates

Integration in polar coordinates can be challenging yet rewarding.The formula \(A = \frac{1}{2}\int_{a}^{b} [f(\theta)^2 - g(\theta)^2] d\theta\) highlights this process.
Here, \(f(\theta)\) and \(g(\theta)\) represent the radial equations of the curves.This equation helps find the area of a region contained between two polar curves, based on their intersection bounds \(a\) and \(b\).

• It's crucial to set up the correct limits. These are found where the two curves intersect.
• Simplify the integrand before carrying out the integration to ease solving.

The integration requires solving for the rotated areas defined by the squared equations.Such calculations provide key insights into the geometric shape features and spatial relationships between curves.