Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Recall the formula \(A=\frac{1}{2} r^{2} \theta\) for the area of the sector of a circle of radius \(r\) and central angle \(\theta\) radians (Section \(10.7) .\) Use this to obtain the formula $$ A=\frac{r_{1}+r_{2}}{2}\left(r_{2}-r_{1}\right)\left(\theta_{2}-\theta_{1}\right) $$ for the area of the polar rectangle \(\left\\{(r, \theta): r_{1} \leq r \leq r_{2},\right.\), \(\left.\theta_{1} \leq \theta \leq \theta_{2}\right\\}\).

Step-by-Step Solution

Answer

The formula for the polar rectangle is derived by subtracting two sector areas and simplifying using the difference of squares.

Step 1: Understanding the Sector Area Formula

The formula for the area \(A\) of a sector of a circle is \(A=\frac{1}{2} r^{2} \theta\), where \(r\) is the radius and \(\theta\) is the angle in radians. This gives the area for a sector, which is a portion of a circle defined by its radius and arc.

Step 2: Conceptualizing the Polar Rectangle

A polar rectangle is a region defined by two radii, \(r_1\) and \(r_2\), and two angles, \(\theta_1\) and \(\theta_2\). We need to calculate the area enclosed by these boundaries in polar coordinates.

Step 3: Calculate the Area Using Sectors

The area of the polar rectangle can be thought of as the area between two sectors. Calculate the area of the larger sector (with radius \(r_2\) and angle \(\theta_2 - \theta_1\)) and subtract the area of the smaller sector (with radius \(r_1\)).

Step 4: Area of the Larger Sector

The area of the larger sector formed by radius \(r_2\) and angle range \(\theta_2 - \theta_1\) is: \[ A_{ ext{large}} = \frac{1}{2} r_2^2 (\theta_2 - \theta_1) \]

Step 5: Area of the Smaller Sector

The area of the smaller sector formed by radius \(r_1\) and the same angle range \(\theta_2 - \theta_1\) is: \[ A_{ ext{small}} = \frac{1}{2} r_1^2 (\theta_2 - \theta_1) \]

Step 6: Subtract Smaller Sector from Larger Sector

Subtract the area of the smaller sector from the area of the larger sector to find the area of the polar rectangle: \[ A = A_{ ext{large}} - A_{ ext{small}} = \frac{1}{2} r_2^2 (\theta_2 - \theta_1) - \frac{1}{2} r_1^2 (\theta_2 - \theta_1) \] Simplifying yields: \[ A = \frac{1}{2}(r_2^2 - r_1^2)(\theta_2 - \theta_1) \]

Step 7: Use the Difference of Squares

Recognize \((r_2^2 - r_1^2)\) as a difference of squares: \((r_2^2 - r_1^2) = (r_2 - r_1)(r_2 + r_1)\). Substitute this back into the formula: \[ A = \frac{1}{2}(r_2 + r_1)(r_2 - r_1)(\theta_2 - \theta_1) \]

Key Concepts

Sector Area Formula

The sector area formula is essential in understanding how to calculate areas within circles. A sector of a circle is like a slice of pie cut from the circle, defined by its radius and the central angle. This angle is typically given in radians. The formula to find the area of a sector, denoted by \( A \), is \( A = \frac{1}{2} r^2 \theta \). Here, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians. This formula is derived from the proportion of the circle's or the sector's arc length to the full circumference.

Applications of this formula are numerous in geometry and trigonometry, where it helps in problems involving circular tracks, parts of circles, compass bearings, and more. By breaking down a circle into sectors, one can solve intricate problems involving areas, lengths, and relationships between angular measurements.

Polar Rectangle

In polar coordinates, a polar rectangle refers to a specific area bounded by two radii and two angle measures. Unlike Cartesian coordinates that use \( x \) and \( y \) to describe location, polar coordinates use \( r \) (radius) and \( \theta \) (angle). This creates a sector-like shape, often called a polar rectangle.

To visualize it, imagine the space contained between two circles of radii \( r_1 \) and \( r_2 \), and the angles \( \theta_1 \) to \( \theta_2 \). The area of this segment can be calculated by considering the differences in areas of sectors formed by the outer boundary \( r_2 \) and the inner boundary \( r_1 \).

Polar rectangles are frequently used in calculus and geometry to solve complex integration problems, especially when changes in angle and radius over time are considered in applications like radar scanning, signal processing, and navigation.

Difference of Squares

The difference of squares is a fundamental algebraic technique used in simplifying expressions and equations. It is expressed by the identity \( a^2 - b^2 = (a - b)(a + b) \). This identity helps in breaking down complex algebraic structures, making them simpler to handle.

In the context of polar rectangles, this concept arises when calculating areas between different sectors. When the formula \((r_2^2 - r_1^2)\) appears, it simplifies to \((r_2 - r_1)(r_2 + r_1)\) using the difference of squares. This substitution allows us to understand the area difference between two circular sections in a manageable form.

Mastering this concept is useful not just in geometry but in solving quadratic equations, factoring complex polynomials, and optimizing expressions in physics and economics.

Circle Geometry

Circle geometry is a fascinating branch of mathematics that deals with the properties and relations of circles. It includes studies of arcs, sectors, chords, tangents, and more.

A circle is defined by its center and radius, and various formulas derive from these basic parameters. Calculating the area of a circle (\( A = \pi r^2 \)) and the circumference (\( C = 2\pi r \)) are foundational skills. These principles extend to circular segments and sectors, giving insight into more complex geometry like polar rectangles.

Applications of circle geometry range from architectural designs involving circular elements to engineering fields where circular components are critical. Understanding topology, or the properties of space that are preserved under continuous transformations, often employs circle geometry. This subject thus serves as a crucial building block for advanced mathematical application and problem-solving.