Problem 51 - Thomas Calculus

Question

In Exercises \(49-52,\) use a CAS to change the Cartesian integrals into an equivalent polar integral and evaluate the polar integral. Perform the following steps in each exercise. $$ \begin{array}{l}{\text { a. Plot the Cartesian region of integration in the } x y \text { -plane. }} \\ {\text { b. Change each boundary curve of the Cartesian region in part }} \\ {\text { (a) to its polar representation by solving its Cartesian equation for } r \text { and } \theta .}\end{array} $$ $$ \begin{array}{l}{\text { c. Using the results in part (b), plot the polar region of integra- }} \\ {\text { tion in the } r \theta \text { -plane. }} \\\ {\text { d. Change the integrand from Cartesian to polar coordinates. De- }} \\ {\text { termine the limits of integration from your coordinates. De- }} \\ {\text { evaluate the polar integral using the CAS integration utility. }}\end{array} $$ $$ \int_{0}^{1} \int_{-y / 3}^{y / 3} \frac{y}{\sqrt{x^{2}+y^{2}}} d x d y $$

Step-by-Step Solution

Answer

The evaluated polar integral is approximately 1.068.

Step 1: Plot the Cartesian Region

The given Cartesian integral corresponds to the region where\( x \) varies from \(-y/3\) to \( y/3 \) and \( y \) ranges from 0 to 1. This region can be visualized as a triangle in the first quadrant of the \( xy \)-plane with vertices at \((0,0)\), \((1/3,1)\), and \((-1/3,1)\). Plot this region in the \( xy \)-plane.

Step 2: Change to Polar Coordinates

To convert Cartesian coordinates to polar coordinates, recall that \( x = r \cos \theta \) and \( y = r \sin \theta \). For the boundary \( x = -y/3 \), sub \( x = r \cos \theta \) and \( y = r \sin \theta \) to get \( r \cos \theta = -r \sin \theta / 3 \), which simplifies to \( \tan \theta = -3 \). This gives \( \theta = \arctan(-3) \). Similarly, for \( x = y/3 \), solve \( \tan \theta = 3 \) to get \( \theta = \arctan(3) \). The curve \( y = 1 \) corresponds to \( r \sin \theta = 1 \), or \( r = 1/\sin \theta \).

Step 3: Plot the Polar Region

Using the boundary equations, the region of integration in polar coordinates is determined by \( \arctan(-3) \leq \theta \leq \arctan(3) \) and \( 0 \leq r \leq 1/\sin \theta \). Plot this region in the \( r\theta \)-plane as a sector bounded by these \( \theta \)-limits and \( r \)-limits.

Step 4: Convert the Integrand to Polar Form

The integrand \( \frac{y}{\sqrt{x^2+y^2}} \) becomes \( \frac{r \sin \theta}{r} = \sin \theta \) in polar coordinates. Thus, the polar integral is \( \int \int \sin \theta \cdot r \, dr \, d\theta \).

Step 5: Determine Integration Limits and Evaluate the Integral

The limits for \( \theta \) are from \( \arctan(-3) \) to \( \arctan(3) \), and \( r \) goes from 0 to \( 1/\sin \theta \). The new integral is \[ \int_{\arctan(-3)}^{\arctan(3)} \int_{0}^{1/\sin \theta} \sin \theta \cdot r \, dr \, d\theta. \] Evaluate using a CAS to find the integral's value.

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a system for specifying coordinates using rectangular grids. In this system, each point in space is represented by an ordered pair

• x-coordinate: distance along the horizontal axis (left-right movement)
• y-coordinate: distance along the vertical axis (up-down movement)

The original exercise involves plotting the Cartesian region of integration, which essentially means depicting a specific area in the xy-plane. For this example, the integration limits for the Cartesian integral define a triangular region within the bounds specified for x and y variables. Understanding this region is foundational before performing coordinate transformations.

Integration Limits

Integration limits in the problem refer to the boundaries of the region over which we integrate. In a Cartesian system, these limits are defined in terms of the variables x and y. For our task, the region is a triangle with vertices

• (0,0)
• (1/3,1)
• (-1/3,1)

In polar coordinates, the integration limits change. Instead of specific x and y bounds, they become r (radius) and θ (angle) limits. To determine these, you need boundaries in angular terms (found in Step 2), and radial terms, which often go from the origin outward to the bounds specified by the problem. This change simplifies the integration process when dealing with circular-related regions.

CAS (Computer Algebra System)

A Computer Algebra System is a type of software that facilitates symbolic mathematics, allowing users to perform algebraic manipulations, solve equations, and evaluate integrals symbolically. CAS is especially useful in this exercise as it helps compute the value of the integral after changing from Cartesian to polar coordinates. With a CAS, you needn't manually solve complex integrals, which can be prone to mistakes or require extensive computation time. Instead, you input the new integral, defined in polar limits, and the system computes it, significantly streamlining the mathematical process.

Coordinate Transformation

Coordinate transformation involves converting a point or a region's description from one coordinate system to another, such as from Cartesian to polar coordinates. This step is crucial in our task because industries use numerous coordinate systems tailored to specific calculations or visualizations.
For the exercise at hand, transforming each Cartesian boundary into polar form starts with utilizing the relationships

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

Using these relationships involves solving for \( r \) and \( \theta \) in terms of \( x \) and \( y \). Once transformed, the polar coordinates often allow for easier integration, especially in cases involving circular symmetry, as they align with the problem's natural geometry.