Problem 49 - Calculus an Applied Approach

Question

Use a symbolic integration utility to evaluate the double integral. $$ \int_{1}^{2} \int_{0}^{x} e^{x y} d y d x $$

Step-by-Step Solution

Answer

The evaluated double integral is \( \int_{1}^{2} e^{x^2} dx - 1 \). This integral cannot be expressed in terms of elementary functions.

Step 1: Solve Inner Integral

Firstly, the inner integral (denoted by 'I') with respect to 'y' should be solved from 0 to 'x' for \( e^{xy} \). Rewrite 'I' as:\[I=\int_{0}^{x} e^{xy} dy\]We perform this integral by substituting \( u = x y \). Then, \( du = x dy \) or \( dy = du/x \). The limits change from 'x=0' to 'x^2'. Therefore, 'I' becomes:\[I= e^{u} \Big|^x_0 = e^{x^2} - e^0 = (e^{x^2} - 1)\]

Step 2: Solve Outer Integral

The function obtained from the first step, \( e^{x^2} - 1 \), is used as the integrand to solve the outer integral (with respect to x):\[J=\int_{1}^{2} (e^{x^2} - 1) dx\]The integral of \( e^{x^2} \) is not standard and does not have a simple antiderivative. Stick to the original form, and the integral of constant '1' from '1' to '2' is '2-1=1'. Finally, the value of double integral is:\[\int_{1}^{2} \int_{0}^{x} e^{x y} d y d x = J = \int_{1}^{2} e^{x^2} dx - 1\]

Key Concepts

Symbolic Integration

Symbolic integration is the process of finding the anti-derivative of a given mathematical function in terms of symbols rather than numerical answers. Think of it as solving a puzzle where you know the patterns (the function) and you're looking for the original picture (the anti-derivative).

For example, in our exercise, the symbolic integration involves finding a function that, when differentiated, results in the given integrand, such as integrating the exponential function pertaining to variable 'x'. The challenge often lies in the complexity of the function and the need to use various integration techniques to find a symbolic solution.

Unlike numerical integration, which approximates the integral's value, symbolic integration provides a formula as a result. This method is highly valued for its exactness and can be used to derive further mathematical information.

Integration Techniques

Integration techniques are the various methods used to evaluate integrals, especially when the function to be integrated is complex. Several techniques, like substitution, integration by parts, and partial fraction decomposition, are employed to simplify these functions and find their integrals.

Examples of Integration Techniques

• Substitution: Also known as the u-substitution, where a part of the integrand is replaced with a single variable, simplifying the integral.
• Integration by Parts: Useful when the integral is a product of two functions.
• Partial Fractions: This breaks down complex rational expressions into simpler fractions before integrating.
• Trigonometric Integrals: Used when an integrand contains trigonometric functions.

In our exercise, the u-substitution method is used for the inner integral, transforming the variable 'y' into 'u', allowing for a solvable integral with the new limits of integration.

U-Substitution Method

The u-substitution method is a technique for evaluating integrals that is especially useful when dealing with composite functions. The idea is to simplify the integrand by substituting a portion of it with a new variable, often dubbed 'u'. This tactic often transforms an intricate integral into a simpler one.

Here's how it works in the context of our exercise:

1. Identify the substitution: Looking at the inner integral, we notice that 'xy' can be treated as a single variable. We call it 'u', hence 'u = xy'.
2. Change of differential: We find 'du' by differentiating 'u'. Since 'u = xy', we have 'du = x dy'.
3. Amend the limits of integration: As 'y' changes from 0 to 'x', 'u' will change from '0' to 'x^2'.
4. Perform the integration: Substitute 'y' with 'u/x' and 'dy' with 'du/x', then integrate with respect to 'u'. The result is a simpler integral to solve.

Ultimately, u-substitution is an effective strategy for finding an antiderivative when direct methods aren't readily available.