Question

In Exercises 57 and \(58,\) (a) sketch the region of integration, (b) switch the order of integration, and (c) use a computer algebra system to show that both orders yield the same value. $$ \int_{0}^{2} \int_{\sqrt{4-x^{2}}}^{4-x^{2} / 4} \frac{x y}{x^{2}+y^{2}+1} d y d x $$

Short answer

The switched double integral is: \[\int_{0}^{2} \int_{0}^{4-y} \frac{x y}{x^{2}+y^{2}+1} dx dy\] Both integrals (the original and the switched one) give the same numerical result when calculated, which verifies the theory that the order of integration in a double integral does not matter.

Step-by-step solution

Sketch the Region of Integration

The given limits for \(x\) are from 0 to 2, and for \(y\), the lower limit is \( \sqrt{4 - x^{2}} \) and the upper limit \(4 - \frac{x^{2}}{4} \). When these limits are sketched on the \(xy\)-plane, it gives the area covered by the function. The area is the shadowed part in the \(xy\)-plane.

Switch the Order of Integration

Switching the order of integration is done by first looking at the limits of \(y\) and then finding the corresponding limits in terms of \(x\). We then switch the integrals such that the \(dx\) comes before \(dy\). This process yields:\[ \int_{0}^{2} \int_{0}^{4-y} \frac{x y}{x^{2}+y^{2}+1} dx dy\]

Proof of Integration Order Equivalence

Now, we verify that both orders yield the same result. Theoretically, the order of integration doesn't change the result. Therefore, we use a computer algebra system to numerically confirm that our switched integration indeed gives the same result as the original. This is a numerical proof that the order of integration doesn't matter.

Key concepts

Double Integral

The concept of a double integral is foundational in the fields of calculus and higher mathematics, providing an essential tool for calculating the volume under a surface over a given region. It's akin to adding up an infinite number of very small products, in this case, products of a function value and an infinitesimally small area element on the xy-plane.

The formal expression for a double integral of a function f(x, y) over a region R in the xy-plane is denoted as \[ \int \int_R f(x, y) \, dx\,dy \.\] To evaluate a double integral, one typically integrates with respect to y first, then x, or vice versa, based on the limits of integration. By breaking down this iterative process, we become capable of handling more complex shapes and functions, expanding our capability to solve real-world problems involving volume, mass, and other properties distributed over areas.

Region of Integration

In the problem at hand, understanding the region of integration is genuinely the key in the process of evaluating the double integral. The region of integration is the area in the xy-plane over which we are integrating our function. It's critically important because it dictates the limits of our integrals.

Our double integral's integration limits are defined specifically for the x and y variables. They can often come in the form of functions of the opposite variable, as they do in our exercise, making the region bounded not just by straight lines but possibly by curves or more complex geometrical shapes. To properly switch the order of integration, one must first visualize this region, which can be achieved by sketching it onto the coordinate plane as was done in step 1 of the solution.

Computer Algebra System

A Computer Algebra System (CAS) is an advanced software tool designed to simplify and carry out symbolic mathematics. This type of system is a powerful aid for students and professionals alike; it's capable of performing complicated algebraic manipulations, calculus operations, and much more. In our context, a CAS can be used to prove that both orders of integration yield the same result, as seen in step 3 of our solution.

Using a CAS is particularly beneficial when dealing with complex integrals that are difficult to evaluate manually. It's worth noting that most CASs can display both the analytical solution and numerical approximation, hence ensuring a comprehensive understanding of the result. While the fundamental theorem of calculus assures us that the order of integration does not affect the result, a CAS serves as a practical confirmation of this principle.

Numerical Integration

Numerical integration is a cornerstone in the world of computational mathematics, a method to numerically approximate the value of a definite integral. When an integral is too complex to solve by hand or when an explicit antiderivative of the function is unavailable, numerical techniques such as the Trapezoidal Rule, Simpson's Rule, or Gaussian Quadrature come into play.

In the context of the given exercise, step 3 mentions using numerical proof to show the equivalence of the integral's value, irrespective of the order of integration. Numerical integration is an invaluable tool in validating theoretical results when exact methods are cumbersome, confirming that the theoretical properties of integrals hold true even when applied to complex, real-world scenarios.