Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 graph is a circle with radius \( a \), and its area is \( \\pi a^2 \).

Step 1: Identify the Equation Form

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

Step 2: Graph Interpretation

Interpret the equation \( r = a \). In polar coordinates, this equation represents all points that are a fixed distance \( a \) from the origin. The locus of these points forms a circle.

Step 3: Draw the Circle

Sketch a circle on the coordinate plane with the origin as the center. The radius of the circle is \( a \), so ensure that all points on the circle are equidistant from the origin by distance \( a \).

Step 4: Calculate the Area

Since a circle is formed, use the standard formula for the area of a circle, \( A = \pi r^2 \). Here, \( r = a \), so the area \( A = \pi a^2 \).

Key Concepts

Circle Equation

In mathematics, understanding the equation of a circle is crucial, especially when we delve into polar coordinates. The equation provided in the exercise, \(r = a\), is a typical form used in polar systems. Unlike Cartesian coordinates, where circles are described using \((x-h)^2 + (y-k)^2 = r^2\), the polar form succinctly states that all points on the circle maintain a constant distance, \(a\), from the origin (the pole). This simplicity comes from the inherent nature of polar coordinates to measure distance and angle from a fixed point.
An advantage of using polar coordinates is that it provides a more intuitive way to describe rotations and circular structures. This is particularly evident as \(r = a\) clearly and directly gives us information about the radius of the circle centered at the origin, making the task of graphing circles straightforward.

Graph Interpretation

Interpreting graphs in polar coordinates can initially seem intimidating, but it becomes more manageable once the foundational concepts are understood. In polar coordinates, the graph of \(r = a\) isn't just a meaningless equation but a representation of a circle with radius \(a\) centered at the origin. Essentially, every point that satisfies \(r = a\) lies on this circle.
Let's break it down further:

• \(r\) indicates the radial distance from the origin to the point in question.
• All points at this constant distance \(a\) effectively "trace" out a circle.
• The circle's center is the pole (or origin) of the polar coordinate system.

By visualizing the radial symmetry inherent in polar coordinates, graphing circles like this becomes more intuitive. This symmetry highlights how each point on the polar plot is equidistant from the central point, making the circle's graph a simple but informative depiction of the equation \(r = a\).

Area Calculation

To find the area enclosed by the circle defined by the polar equation \(r = a\), we utilize the familiar formula for the area of a circle: \(A = \pi r^2\). Since in our situation \(r = a\), the area becomes \(A = \pi a^2\).
Calculating this area is straightforward once we have pinpointed what \(a\) represents — the radius of the circle. This formula emerges from polar and Cartesian considerations and is useful in numerous applications ranging from geometry to physics.
Some key points to consider during area calculation include:

• The constant \(\pi\) represents the ratio of the circumference of any circle to its diameter.
• The radius \(a\) dictates the size — larger \(a\) implies a more expansive circle.
• Understanding that the formula \(\pi a^2\) directly stems from the circle's properties helps demystify calculations.

Overall, area calculation in polar coordinates using this circle equation is efficient, enabling quick and exact determinations of the region encased by the curve.