Problem 75 - University Calculus: Early Transcendentals

Question

Circular sector Integrate \(f(x, y)=\sqrt{4-x^{2}}\) over the smaller sector cut from the disk \(x^{2}+y^{2} \leq 4\) by the rays \(\theta=\pi / 6\) and \(\theta=\pi / 2\)

Step-by-Step Solution

Answer

Integrate using polar coordinates, evaluate inner and outer integrals.

Step 1: Convert to Polar Coordinates

To solve the integration problem, convert the given function and region from Cartesian coordinates to polar coordinates. In the polar coordinate system, let \( x = r \cos \theta \) and \( y = r \sin \theta \). The differential area element becomes \( r \, dr \, d\theta \). For the given disk \( x^2 + y^2 \leq 4 \), the radius \( r \) ranges from 0 to 2.

Step 2: Define the Region of Integration

The smaller sector of the disk is bounded by the angles \( \theta = \pi / 6 \) and \( \theta = \pi / 2 \). Thus, the integral's limits for \( \theta \) will be from \( \pi / 6 \) to \( \pi / 2 \), and for \( r \) from 0 to 2.

Step 3: Set Up the Integral

The integrand \( f(x, y) = \sqrt{4 - x^2} \) converts to \( \sqrt{4 - r^2 \cos^2 \theta} \) in polar coordinates. Set up the double integral for the area:\[ \int_{\pi/6}^{\pi/2} \int_{0}^{2} \sqrt{4 - r^2 \cos^2 \theta} \times r \; dr \; d\theta \]

Step 4: Evaluate the Inner Integral

First, evaluate the integral with respect to \( r \):\[ \int_{0}^{2} r \sqrt{4 - r^2 \cos^2 \theta} \; dr \]This part involves a substitution. Let \( u = 4 - r^2 \cos^2 \theta \), then \( du = -2r \cos^2 \theta \, dr \). After substituting and simplifying, evaluate the definite integral from \( r = 0 \) to \( r = 2 \).

Step 5: Evaluate the Outer Integral

After computing the integral with respect to \( r \), evaluate the resulting expression with respect to \( \theta \) from \( \pi/6 \) to \( \pi/2 \). This step involves integrating a function of \( \theta \) that results from evaluating the inner integral.

Step 6: Compute the Final Value

After completing the integration for both variables, calculate the numerical value to find the area of the sector. This results in the final answer for the integral over the specified region.

Key Concepts

Double Integration

Double integration is a mathematical process used to compute the volume or area under a surface. In this context, it helps to find the area of the sector of a circle. When dealing with double integrals, you usually integrate a function with respect to one variable first, and then with respect to the second variable.
This involves setting up an integral with two differentials, such as \(\int \int f(x, y) \, dx \, dy\). The order of integration can sometimes be interchanged provided that the limits allow it, but in this problem, we respect the limits for polar coordinates.

• Inner integral: Integrate with respect to one variable, keeping the other constant.
• Outer integral: Integrate the result of the inner integral with respect to the other variable.

This structured approach will yield an exact total area or volume under the given domain.

Cartesian Coordinates

Cartesian Coordinates provide a framework where each point is identified by an ordered pair of numbers, typically denoted as \( (x, y) \). These coordinates are crucial when dealing with integration because they describe the location of points on a two-dimensional plane.
In mathematics, problems and equations are often based in the Cartesian system, where plotting requires axes perpendicular to each other.

• X-coordinate determines the horizontal position.
• Y-coordinate indicates the vertical position.

In the given exercise, the function \( \sqrt{4 - x^2} \) is originally determined in this system before switching to polar coordinates to facilitate the integration over the defined sector area.

Integration Limits

Integration limits define the boundaries over which a function is to be evaluated. These limits are essential in solving double integrals as they pinpoint the start and end values for both variables.
In this problem, the region over which you integrate is a sector of a circle limited by polar angles and radius.

• The \( \theta \) limits are the angles given as \( \pi / 6 \) to \( \pi / 2 \).
• The \( r \) limits are the radial distance from the center, here from 0 to 2, indicating the circle's radius.

Each limit corresponds to a specific variable, and it is crucial to apply these correctly to encapsulate the desired geometrical area or volume in your integral.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrations, especially when dealing with radicals involving quadratic expressions. In this exercise, a substitution method was necessary to accurately evaluate the integral because of the form \( \sqrt{4 - r^2 \cos^2 \theta} \).
When you encounter such expressions, you can use trigonometric identities or substitutions to transform them into more integrable forms.

• For example: Replace terms like \( \sqrt{a^2 - x^2} \) by substituting \( x = a \sin \theta \) or similar.
• It often leads to an easier integration process by transforming the expression into a more recognizable pattern.

To complete the integration in the original exercise, such substitutions help manage the integrals, leading to a solution that's easier to compute and interpret.