Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each of the iterated integrals. $$ \int_{0}^{\pi / 2} \int_{0}^{1} x \sin x y d y d x $$

Step-by-Step Solution

Answer

The solution setup involves complex integration, often resolved using numerical methods or special functions.

Step 1: Integrate with respect to y

To evaluate the iterated integral \( \int_{0}^{\pi / 2} \int_{0}^{1} x \sin(xy) \, dy \, dx \), we first hold \( x \) constant and integrate with respect to \( y \). The integral becomes \( \int_{0}^{1} x \sin(xy) \, dy \). To solve this, use the substitution \( u = xy \), which gives \( du = x \, dy \) or \( dy = \frac{du}{x} \). The limits of \( y \) from 0 to 1 transform \( u \) from 0 to \( x \). The integral becomes \( \int_{0}^{x} \sin(u) \frac{1}{x} \, du \). This simplifies to \( \frac{1}{x} \left[ -\cos(u) \right]_{0}^{x} \). Substitute back to get \( \frac{1}{x}(-\cos(x) + 1) = \frac{1 - \cos(x)}{x} \).

Step 2: Integrate with respect to x

Now integrate with respect to \( x \) over the interval from 0 to \( \pi/2 \): \( \int_{0}^{\pi/2} \frac{1 - \cos(x)}{x} \, dx \). This is a challenging integral to evaluate analytically without special functions, so we must focus on determining it as a conceptual step in the setup of iterated integrals. However, we note that integration by parts or numerical techniques could give an approximate value. The precise integration of this form would ideally use advanced mathematical techniques or numerical methods for a solution.

Key Concepts

Integration Techniques

In calculus, integration techniques are methods used to find the integral of functions. These techniques are crucial because they allow us to compute areas under curves, volumes, and solve various problems in physics and engineering.
There are several common integration techniques:

• Basic Integration: Involves straightforward integration of functions that have simple antiderivatives.
• Substitution: Used when a direct integration is not possible and involves substituting parts of the integral with a new variable.
• Integration by Parts: Useful for integrating products of functions.
• Partial Fractions: Used for integrating rational functions by breaking them into simpler fractions.

Each of these techniques can solve specific types of integrals, and often, multiple techniques are combined to solve more complex ones.
In the given exercise, we applied substitution and recognized that future techniques could assist in completing the problem.

Substitution Method

The substitution method is a powerful technique to simplify an integral. It works by transforming the variable of integration into a new variable, making the integral easier to compute.
In the exercise, we used substitution by setting up a new variable, \( u = xy \), which leads to derivative transformation \( du = x \, dy \). By rewriting \( dy \) as \( \frac{du}{x} \), the integral simplifies, enabling easier integration over the limits.
This method is particularly helpful when an integral contains products or compositions of functions, such as trigonometric or exponential functions. It simplifies the function inside the integral and often resolves complex integrals into standard form.

Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation and is particularly useful for integrating products of functions.
It is expressed as \( \int u \, dv = uv - \int v \, du \). Here, \( u \) and \( dv \) are parts carefully chosen from the integral. This transformation sometimes simplifies the integration process significantly.
In our example, integration by parts may help in solving \( \int_{0}^{\pi/2} \frac{1 - \cos(x)}{x} \, dx \) effectively. While straightforward calculations are complex, this method leverages the derivative and integral connections, facilitating simplification by recursive application. Often, identifying the right functions for \( u \) and \( dv \) is key to reducing the integral into simpler components.

Numerical Integration

Sometimes, an integral cannot be solved analytically, and numerical integration methods become necessary. These techniques approximate the value of an integral using discrete data points.
Common numerical methods include:

• Trapezoidal Rule: Approximates the region under the curve as a series of trapezoids.
• Simpson's Rule: Uses parabolic segments instead of linear ones for approximation.
• Monte Carlo Integration: Useful for multidimensional integrals and is based on randomness.

In situations like the given integral, \( \int_{0}^{\pi/2} \frac{1 - \cos(x)}{x} \, dx \), when analytical methods become too taxing or impossible, these numerical approaches provide approximate solutions. They are invaluable in engineering and sciences, where exact solutions might be less crucial than obtaining a workable approximation quickly.