Problem 47 - College Algebra and Calculus: An Applied Approach

Question

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

The value of the double integral is \(\frac{1}{2} e^{-1} (e^{4}-1)-\frac{1}{2}(e^{4}-1)\)

Step 1: Identification

Identify the inner and outer integrals. The integral with respect to \(x\) is the inner integral and the one with respect to \(y\) is the outer integral. Here, the inner integral is \(\int_{0}^{2} e^{-x^{2}-y^{2}} dx\) and the outer one is \(\int_{0}^{1}\).

Step 2: Compute the inner integral

The inner integral should be computed first. The limit of \(x\) goes from 0 to 2. This can be found using a symbolic integration calculator, to get the integral as \(-\frac{1}{2} e^{-4+y^{2}}(e^{4}-1)\).

Step 3: Compute the outer integral

Now, the outer integral can be computed. Substitute the computed value of the inner integral into the outer integral. This can also be found using a symbolic integration calculator, to get the integral as \(\frac{1}{2} e^{-1} (e^{4}-1)-\frac{1}{2}(e^{4}-1)\).

Key Concepts

Symbolic Integration

Symbolic integration is an essential tool in calculus, allowing us to find the exact antiderivative of a function. Unlike numerical integration, which provides an approximate value, symbolic integration yields a precise expression that encompasses the indefinite integral of a given function. In the case of the exercise, the function to be integrated is \( e^{-x^{2}-y^{2}} \), which presents an exponential function involving the variables \( x \) and \( y \) raised to the power of \( -1 \). Symbolic integration is especially useful when dealing with complex functions that might not be easily integrated by standard methods, such as parts or substitution, often requiring the help of sophisticated algorithms - the kind that can be found in many computer algebra systems today.

When approaching such an integral symbolically, one typically makes use of fundamental integration rules and techniques tailored to the specific type of function. This process involves identifying patterns within the integrand that match known integral forms. In an educational setting, mastering these techniques allows for greater understanding of the underlying mathematics and the ability to solve a wide range of problems.

Inner and Outer Integrals

In a double integral, the concept of inner and outer integrals is pivotal. The distinction lies in the order of integration - the inner integral is computed first, followed by the outer one.

Understanding the Sequence

In our example, \( \int_{0}^{1} \int_{0}^{2} e^{-x^{2}-y^{2}} dx dy \), the integration with respect to \( x \) (from 0 to 2) is the inner integral, and the integration with respect to \( y \) (from 0 to 1) is the outer integral. Processing them in the correct order is crucial: the inner integral acts as an input for the outer one.

Why Order Matters

This order can transform the complex two-variable expression into a simpler form that only involves the outer variable, as we integrate and replace the inner variable with its bounds. Thus, understanding and correctly identifying inner and outer integrals simplifies the process of evaluating double integrals.

Integration Limits

Setting appropriate integration limits is critical when solving integrals, as they define the boundaries within which the integral is evaluated. In double integrals like \( \int_{0}^{1} \int_{0}^{2} e^{-x^{2}-y^{2}} dx dy \), the limits are respectively 0 to 2 for \( x \) and 0 to 1 for \( y \).

• The limits for \( x \) are known as 'inner limits' and those for \( y \) as 'outer limits'.
• These limits can represent the boundaries of a region in the \( xy \) plane over which the function is to be integrated.

Students must pay careful attention to these limits since incorrectly setting them can lead to inaccurate results. In problems involving complex regions or variable limits, sketching the region or using a graph can greatly help in determining the correct bounds for integration.

Symbolic Integration Calculator

A symbolic integration calculator is an advanced tool that assists in performing integrations symbolically, as demonstrated in the step-by-step solution to our exercise. These calculators leverage algorithms to process and simplify mathematical expressions, especially in cases where manual computation is tedious or challenging.

Advantages of Using Calculators

Such tools can handle a variety of integrals, from simple polynomials to more complex functions involving exponentials, trigonometry, and beyond. They are particularly useful for verifying manual calculations or for exploring the properties of integrals that are not straightforward to solve by hand.

Choosing the Right Tool

While several free and subscription-based symbolic integration calculators are available online, the choice of tool might depend on the complexity of the function, the user interface, and the level of detail provided in the solution. For students, these calculators can serve as a method to check work and foster a deeper understanding by comparing manual and computed solutions.