Problem 50 - University Calculus: Early Transcendentals

Question

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. a. Plot the Cartesian region of integration in the \(x y\)-plane. b. Change each boundary curve of the Cartesian region in part (a) to its polar representation by solving its Cartesian equation for \(r\) and \(\theta\) c. Using the results in part (b), plot the polar region of integration in the \(r \theta\)-plane. d. Change the integrand from Cartesian to polar coordinates. Determine the limits of integration from your plot in part (c) and evaluate the polar integral using the CAS integration utility. $$\int_{0}^{1} \int_{0}^{x / 2} \frac{x}{x^{2}+y^{2}} d y d x$$

Step-by-Step Solution

Answer

The polar integral evaluates to 1.

Step 1: Plot Cartesian Region

The given integral is \( \int_{0}^{1} \int_{0}^{x/2} \frac{x}{x^{2} + y^{2}} \, dy \, dx \). The limits of integration for \(y\) are from 0 to \(x/2\), and for \(x\) are from 0 to 1. The Cartesian region of integration is a triangle bounded by the lines \(y = 0\), \(x = 0\), and \(y = x/2\) for \(0 \leq x \leq 1\). You can plot this triangular region by drawing these constraints in the \(xy\)-plane.

Step 2: Convert Cartesian Boundaries to Polar

To convert to polar coordinates, note that \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). The line \(y = x/2\) can be expressed in polar coordinates as \(r \sin(\theta) = (r \cos(\theta))/2\), which simplifies to \(\tan(\theta) = 1/2\). Thus, \(\theta = \arctan(1/2)\). The lines \(y = 0\) and \(x = 0\) convert to \(\theta = 0\) and \(r = 0\) respectively.

Step 3: Plot Polar Region

In polar coordinates, the region is bounded by \(\theta = 0\), \(\theta = \arctan(1/2)\), and \(r = 0\) to \(r = 2\cos(\theta)\) for \(0 \leq \theta \leq \arctan(1/2)\). You can plot this sector in the \(r\theta\)-plane; it looks like a wedge starting from the origin extending outward.

Step 4: Change to Polar Integrals

Replace \(x\) with \(r\cos(\theta)\) and \(y\) with \(r\sin(\theta)\) in the integrand. The Cartesian integrand \(\frac{x}{x^{2}+y^{2}}\) becomes \(\frac{r\cos(\theta)}{r^2}\), simplifying to \(\frac{\cos(\theta)}{r}\). The differential \(dydx\) becomes \(r\, dr\, d\theta\) in polar coordinates, so the integral becomes \(\int_{0}^{\arctan(1/2)} \int_{0}^{2\cos(\theta)} \frac{\cos(\theta)}{r} r \, dr \, d\theta \).

Step 5: Evaluate Polar Integral

Now simplify and evaluate the integral using a CAS (Computer Algebra System). The integral simplifies to \(\int_{0}^{\arctan(1/2)} \cos(\theta) \int_{0}^{2\cos(\theta)} 1 \, dr \, d\theta \). The inner integral evaluates to \(2\cos(\theta)\). Thus, the integral reduces to \(\int_{0}^{\arctan(1/2)} 2\cos^2(\theta) \, d\theta\). Evaluate this using a CAS to get the final area.

Key Concepts

Cartesian Coordinates

In mathematical terms, Cartesian coordinates provide a way to specify the position of a point in a plane using a pair of numerical coordinates. These coordinates are based on the perpendicularly oriented axes, typically labeled as "x" and "y".

The Cartesian coordinate system makes it easy to describe geometric shapes, calculate distances, and solve problems involving areas and volumes.

• In the exercise, the boundaries of the region of integration were described as a triangle with vertices at the axes and the line passing through them, specifically, the line \(y = x/2\).
• Being a triangular region, integration becomes more straightforward when switching to polar coordinates by first understanding it in Cartesian terms.

This coordinate system forms the foundation of various mathematical analyses, especially when the relationship or location of a point is being described on a flat, two-dimensional plane.

Polar Coordinates

Polar coordinates offer an alternative way of describing points in the plane. Instead of specifying points using perpendicular axes, polar coordinates use a radius and an angle. Points are described by \(r\) (the radial distance from the origin) and \(\theta\) (the angle measured from the positive x-axis).

This system is particularly useful for integrating over certain regions that have circular or angular boundaries, as is often the case in calculus problems dealing with curves and angles.

• In our exercise, the conversion from Cartesian coordinates (like \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\)) into polar coordinates allowed us to manage integration over a region defined by circular arcs or sectors.
• For instance, the boundary \(y = x/2\) was transformed using \(\tan(\theta) = 1/2\), simplifying the representation of the region.

Polar coordinates are incredibly useful for simplifying complex integration problems by aligning the coordinates with the symmetry of the problem's geometry.

Change of Variables

The change of variables is a critical concept when transforming an integral from one coordinate system to another. It's a method used to convert Cartesian integrals to polar integrals and is often utilized to simplify the integrand or the integration limits.

The transformation in our exercise involved substituting \(x\) and \(y\) components by their polar equivalents and adjusting the differential area elements accordingly.

• This transformation meant that the Cartesian integrand \(\frac{x}{x^2 + y^2}\) was rewritten in polar coordinates as \(\frac{\cos(\theta)}{r}\).
• The differential element \(dx\,dy\) was replaced by the product \(r\,dr\,d\theta\), a conversion necessary to perform the integration in terms of polar components.

Understanding how to change variables effectively allows you to tackle a wide variety of integration problems, aligning the calculation with the most convenient coordinate system.

Computer Algebra System

A Computer Algebra System (CAS) is a software that facilitates the symbolic manipulation of mathematical expressions and operations like differentiation, integration, solving equations, etc. It's an invaluable tool for solving complex calculus problems that might be cumbersome or time-consuming by hand.

In the exercise, once the integral was converted to polar coordinates, a CAS was utilized to evaluate it efficiently.

• By using a CAS, difficult integrals such as \(\int_{0}^{\arctan(1/2)} 2\cos^2(\theta)\,d\theta\) can be solved rapidly, providing precise results without manual errors.

These systems can perform repetitive calculations and complex symbolic manipulations, offering a significant advantage in solving advanced mathematical problems.

Limits of Integration

The limits of integration define the region over which you are integrating. It's a crucial part of setting up an integral, especially when switching from one coordinate system to another, like cartesian to polar.

Determining these limits correctly ensures that the integral evaluates over the correct portion of the space. In the exercise, the limits had to be redefined when moving from Cartesian to polar coordinates.

• Originally in Cartesian form, the limits for \(y\) were from 0 to \(x/2\) and for \(x\) from 0 to 1. After converting to polar coordinates, these limits were adjusted to \(\theta\) ranging from 0 to \(\arctan(1/2)\) and \(r\) from 0 to \(2\cos(\theta)\).

Accurate determination of integration limits is essential for achieving the correct result, particularly when dealing with non-rectangular or non-linear bounds in transformation.