Problem 47 - Calculus an Applied Approach

Question

In Exercises \(47-54,\) use a symbolic integration utility to evaluate the double integral. $$ \int_{0}^{1} \int_{0}^{2} e^{-x^{2}-y^{2}} d x d y $$

Step-by-Step Solution

Answer

Unfortunately, this double integral doesn't have a standard form in the elementary functions. So, it can't be evaluated explicitly. It has to be calculated numerically or symbolically as per the problem statement, and in many such cases, exact solutions are often represented in terms of special functions or approximations. The exact symbolic solution would thus be represented in terms of 'erf', the error function or a numerical approximation would have to be made.

Step 1: Evaluate the Inside Integral

The integral is arranged such that the most nested one should be solved first. The inner integral is with respect to x. Thus, initially ignore the y-components and integrate \(e^{-x^{2}-y^{2}}\) from 0 to 1 with respect to x. This gives \(\int_{0}^{1} e^{-x^{2}-y^{2}} dx\).

Step 2: Evaluate the Outside Integral

Now the inner integral is replaced by its result, and the previously neglected y-components are now considered. The outer integral should now be solved by integrating the result of First Step from 0 to 2 with respect to y. This becomes \(\int_{0}^{2} [Result from Step 1] dy\).

Step 3: Simplify the Result

Lastly, perform the y-integration and simplify the result if possible. The result should be a numerical value as the limits of both the integrals are dynamic.

Key Concepts

Symbolic Integration

Symbolic integration involves finding the integral of a function using algebraic symbols rather than numerical approximations. In the case of the given double integral, \[ \int_{0}^{1} \int_{0}^{2} e^{-x^{2}-y^{2}} dx \, dy \] we employ symbolic integration to express the result in terms of functions and constants. Using symbolic integration helps simplify mathematical expressions and understand the theoretical behaviour of functions.

• Symbolic integration allows us to manipulate expressions, differentiating and integrating symbols rather than numbers.
• It's powerful for theoretical exploration and conveying precise mathematical concepts.
• In some cases, symbolic integration helps identify patterns or generate general solutions.

For the exercise above, symbolic integration might utilize known integral formulas or transformations to handle complex expressions like those involving exponential functions.

Integration by Parts

Integration by parts is a technique used to evaluate integrals where the standard approach doesn't easily apply. It is based on the product rule for differentiation and can be described by the formula: \[ \int u \, dv = uv - \int v \, du \] where \( u \) and \( dv \) need to be chosen strategically. This method could be handy if the inner integral of a double integral like \[ \int_{0}^{1} e^{-x^{2}-y^{2}} dx \] is complex, although in our specific example, it doesn't directly apply as the integrand involves exponential powers of variables.

• Correctly choosing \( u \) and \( dv \) is crucial for simplifying the problem.
• Integration by parts can turn an unsolvable integral into a solvable one.
• It's often used in solving integrals involving product of functions."

Although not used directly in solving the given integral, understanding this technique is beneficial in tackling more complicated integrals that arise during similar exercises.

Iterated Integrals

Iterated integrals break down multi-variable integration problems into a series of single-variable integrals. In our exercise, \[ \int_{0}^{1} \int_{0}^{2} e^{-x^{2}-y^{2}} dx \, dy \] the double integral is solved as two separate integrals, one inside the other, using the limits provided. This process starts with the inner integral, keeping one variable constant, and then proceeds to the outer integral. Here is the step-by-step procedure for iterated integrals:

• Solve the inner integral first by treating the outer variable as a constant.
• Incorporate the result into the outer integral.
• Evaluate the outer integral to find the final solution.

Iterated integrals simplify the process of integrating over a region by breaking it down into manageable parts. They enable the handling of more complex integrals through separation and systematic solving.