Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1–10, sketch the graph of the given equation and find the area of the region bounded by it. $$ r=a, a>0 $$

Step-by-Step Solution

Answer

The area of the region is \( \pi a^2 \).

Step 1: Understanding the Equation

The given equation is in polar coordinates: \( r = a \), where \( a > 0 \). This equation represents a circle with a radius of \( a \) centered at the origin (pole).

Step 2: Sketching the Graph

To sketch the graph of \( r = a \), plot a circle with radius \( a \) centered at the origin on a polar coordinate system. This is because the equation represents all points at a distance \( a \) from the origin.

Step 3: Setting Up the Area Formula

The area \( A \) of a circle in polar coordinates is given by \( A = \frac{1}{2} \int_{0}^{2\pi} r^2 \, d\theta \). For the circle \( r = a \), the formula becomes \( A = \frac{1}{2} \int_{0}^{2\pi} a^2 \, d\theta \).

Step 4: Integrating to Find the Area

Since \( a^2 \) is a constant, the integral simplifies to \( A = \frac{1}{2} a^2 \int_{0}^{2\pi} \! d\theta = \frac{1}{2} a^2 [\theta]_{0}^{2\pi} = \frac{1}{2} a^2 (2\pi - 0) \).

Step 5: Calculating the Final Area

Compute \( A = \pi a^2 \). This is the area of the circle with radius \( a \).

Key Concepts

Circle in Polar Coordinates

In polar coordinates, the equation \( r = a \) describes a circle. This is a simple yet fundamental concept in polar graphing. Let's break down why this is the case. Here, \( r \) represents the radial distance from the origin (also called the pole) to any point on the circle, and \( a \) is a constant greater than zero, representing the radius of the circle.

When plotting on a polar coordinate graph, each point has a distance \( r \) from the origin and an angle \( \theta \) from the positive x-axis. For the circle described by \( r = a \):

• All points have the same distance \( a \) from the origin.
• By varying \( \theta \) from 0 to \( 2\pi \), the entire circle is drawn.

Thus, \( r = a \) forms a circle centered at the origin, with a constant radius \( a \) in a 360-degree sweep around the origin.

Area in Polar Coordinates

To find the area of a region in polar coordinates, we use a specific formula. Generally, the area \( A \) enclosed by a curve from angle \( \theta = \alpha \) to \( \theta = \beta \) is given by:

• \( A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta \)

For our circle \( r = a \), we want the area for one complete rotation, i.e., \( \theta \) from 0 to \( 2\pi \).

Plugging \( r = a \) into the formula gives:

• \( A = \frac{1}{2} \int_{0}^{2\pi} a^2 \, d\theta \)

This simplifies the process as \( a^2 \) is a constant, making the integration straightforward.

Integration in Polar Coordinates

The integration step is crucial to computing the area in polar coordinates, particularly for an equation like \( r = a \). As previously stated, the integral simplifies when \( r \) is a constant.

When evaluating \( A = \frac{1}{2} \int_{0}^{2\pi} a^2 \, d\theta \):

• First, \( a^2 \) being constant is factored out.
• The integral reduces to \( \frac{1}{2} a^2 \int_{0}^{2\pi} d\theta \).
• This becomes \( \frac{1}{2} a^2 [\theta]_{0}^{2\pi} \).
• Finally, calculate the definite integral, \( 2\pi - 0 = 2\pi \), yielding \( \frac{1}{2} a^2 \cdot 2\pi = \pi a^2 \).

This result confirms that the area enclosed by a circle with radius \( a \) is \( \pi a^2 \), as expected.