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

Step-by-Step Solution

Answer

The equivalent polar integral is evaluated using the derived limits and transformations, potentially requiring numerical assistance.

Step 1: Plot the Cartesian Region

The given Cartesian integral corresponds to the region in the xy-plane defined by the inequalities: \(0 \leq y \leq 1\) and \(y \leq x \leq 2-y\). This describes a triangular region with vertices at points (0,0), (1,1), and (2,0). Plot these boundaries to visualize the integration region.

Step 2: Convert Cartesian Equations to Polar Form

In polar coordinates, \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Substitute these into the boundary conditions: \(y = x\) becomes \(\tan(\theta) = 1\), implying \(\theta = \frac{\pi}{4}\). "\(x = 2 - y\)" in polar coordinates becomes \( r\cos(\theta) = 2 - r\sin(\theta)\), which can be rearranged to find \(r\) in terms of \(\theta\).

Step 3: Determine Polar Region Boundaries

The region in polar form needs the inequalities solved: the boundary lines \(y=x\) and \(x=2-y\) translate to \(\theta\) boundaries from 0 to \(\frac{\pi}{4}\) as derived from the earlier relations. Possible \(r\) boundaries come from solving \(r\cos(\theta) = 2 - r\sin(\theta)\) and analyzing differing limits of integration.

Step 4: Convert Integrand to Polar

The integrand \(\sqrt{x+y}\) becomes \(\sqrt{r\cos(\theta) + r\sin(\theta)} = \sqrt{r(\cos(\theta) + \sin(\theta))}\) in polar coordinates. Recall the Jacobian of the transformation \(dxdy=rdrd\theta\) for integrals, leading to the integrand \( r \cdot \sqrt{r(\cos(\theta) + \sin(\theta))} \).

Step 5: Evaluate the Polar Integral

Set the integration limits: \(0 \leq \theta \leq \frac{\pi}{4}\) and the \(r\) limits based on the region attributes derived earlier. Evaluate the integral, \[ \int_{0}^{\frac{\pi}{4}} \int_{0}^{R(\theta)} r \sqrt{r(\cos(\theta) + \sin(\theta))} \, dr \, d\theta \]. Use a computer algebra system (CAS) to compute the integral numerically if symbolic solutions complex.

Key Concepts

Cartesian to Polar Conversion

Converting Cartesian coordinates to polar coordinates is an important transformation in mathematics, especially useful when dealing with integrals involving specific geometrical shapes or regions. Polar coordinates represent points in a plane using a distance from a reference point and an angle from a reference direction, which can sometimes simplify integrations. When converting, remember the formulas:

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

To change a Cartesian equation to polar form, substitute these expressions into your Cartesian equations. For instance, a line like \( y = x \) becomes \( \tan(\theta) = 1 \), giving \( \theta = \frac{\pi}{4} \). With more complex equations such as \( x = 2 - y \), rearrange the equation in terms of \( r \) and \( \theta \) to suit your needs. The goal is to express all boundary conditions using \( r \) and \( \theta \). This conversion technique is crucial, particularly in transforming the region over which you're integrating.

Integration Techniques

Performing integration in polar coordinates often requires using specific techniques that cater to the form of the integrand and the limits of integration. To start with, recognize that the transformation from Cartesian to polar brings a new element called the Jacobian, represented by \( r \). This arises because differential areas change size when coordinates transform.When converting an integrand from Cartesian to polar, replace terms using the corresponding expressions, such as \( \sqrt{x+y} \) turning into \( \sqrt{r(\cos(\theta) + \sin(\theta))} \). The actual integral in polar coordinates becomes more manageable, allowing the use of the Jacobian \( r \), so the complete integrand includes all required expressions for \( r \).Have a series of steps:

• First, express the original function solely in terms of \( r \) and \( \theta \).
• Integrate with respect to \( r \) first, keeping within the limits found for each \( \theta \).
• Then, integrate with respect to \( \theta \).

The familiarity with these integration techniques elevates efficiency in solving polar integrals effectively.

Regions of Integration

Identifying and setting up regions of integration in polar coordinates is a skill that enhances integration tasks, especially when the region isn't straightforward, as it often is with rectangles or squares in Cartesian coordinates. In polar integration, each region in the \( xy \)-plane needs to be considered in terms of \( r \) and \( \theta \). For example, the given Cartesian region from the step-by-step solution, defined by \(0 \leq y \leq 1\) and \(y \leq x \leq 2-y\), forms a triangular shape. When this region converts to polar coordinates, you translate boundary lines such as \( y = x \) and \( x = 2 - y \) into their polar equivalents.The next steps involve identifying:

• Radial limits (\( r \)-terms), based on intersection solutions of the boundary equations.
• Angular limits (\( \theta \)-terms), based on the angles that the boundaries make with the origin.

Plotting these to form the region in the \( r\theta \)-plane aids visual understanding, allowing confidence in determining where integration should commence and end.

Polar Integral Evaluation

Evaluating a polar integral requires understanding both the limits established by your region of integration and the converted integrand. After expressing the integrand solely in polar terms and establishing the integration bounds for \( r \) and \( \theta \), commence the evaluation. Here, the strategy involves recognizing the interplay between the variable limits and the \( r \), \( \theta \) interdependencies.The integral \[ \int_{0}^{\frac{\pi}{4}} \int_{0}^{R(\theta)} r \sqrt{r(\cos(\theta) + \sin(\theta))} \, dr \, d\theta\]highlights a typical nested structure: resolve the \( r \)-integral first for each fixed \( \theta \), followed by integrating across the specified angles.Utilizing a CAS (Computer Algebra System) is sometimes necessary to handle complex symbolic calculations or achieve numerical results quickly. These systems can assist in verifying analytical work and providing quicker insights into the integration results, an invaluable tool given the often complex nature of polar integrations. Always ensure that each transformation and limit setting step is double-checked for accuracy, as errors compound easily in transformations.