Problem 24 - Thomas Calculus

Question

Substitutions in single integrals How can substitutions in single definite integrals be viewed as transformations of regions? What is the Jacobian in such a case? Illustrate with an example.

Step-by-Step Solution

Answer

Substitutions transform the integration domain, scaled by a Jacobian, such as \( \\frac{1}{2}\sec^2(x) \) in the example.

Step 1: Understanding Substitution in Single Integrals

Substitution in single integrals involves changing the variable of integration to simplify the integral. This is similar to a transformation of the domain of the function we are integrating, mapping the original region to a new one. Such substitutions often take the form of a change of variable, such as from \( x \) to \( u \) where \( u = g(x) \).

Step 2: Introducing the Jacobian in Integrals

The Jacobian in this context is the derivative of the substitution function, \( g(x) \). When substituting \( u = g(x) \), the differential changes by the factor of \( g'(x) \). For definite integrals, this means the integral \( \int_a^b f(x) \, dx \) under substitution becomes \( \int_{g(a)}^{g(b)} f(g^{-1}(u)) \cdot \left| (d/du)(g^{-1}(u)) \right| \, du \).

Step 3: Applying Substitution to an Example

Consider changing variable in the integral \( \int_0^rac{ an(1)}{2} \frac{2}{1+u^2} \, du \) to \( x = \tan^{-1}(2u) \). Here, \( dx = \frac{2}{1+(2u)^2} \, du \). Thus, \( u = \frac{1}{2}\tan(x) \) and \( du = \frac{1}{2}\sec^2(x) \, dx \). After substitution and simplification, the integral becomes \( \int_0^1 \, dx \), which evaluates to \( 1 \).

Step 4: Evaluating the Effect of Jacobian

The Jacobian \( \frac{1}{2}\sec^2(x) \) reflects the stretching effect of the substitution on the original region. This scaling factor ensures that the area under the curve in the new variable \( x \) matches the original area under the transformation.

Key Concepts

Transformation of Regions

When performing substitutions in integrals, we essentially transform the region from one set of bounds to another. Think of it this way: every integral computes the area under a curve within certain limits. By changing variables, you're mapping the original region, defined by the initial bounds, to a new one, which might be easier to understand or compute. For instance, suppose you're integrating over a difficult interval from \( x \) to \( u \) by setting \( u = g(x) \).

• This acts like a mathematical lens, zooming in or out of certain parts of the graph.
• It helps simplify or linearize complex regions, making calculations less cumbersome.

By choosing an appropriate substitution, you control how the region of integration deforms. This transformation can significantly simplify integral evaluation, transforming a complicated function into a more manageable form.

Jacobian

The Jacobian comes into play as the scaling factor when we switch from one variable to another during integration. Imagine it as the tool that adjusts the scale of measurement between your original function and the transformed function.

• The Jacobian for a single variable transformation is \( g'(x) \), the derivative of your substitution function.
• It modifies the differential element \( dx \) in the integral to account for the transformation.
• Using \( g'(x) \) ensures the area calculated respects how the function stretches or compresses the interval of integration.

So, when integrating from \( a \) to \( b \) in terms of \( x \), and transforming to \( u \), the integral transforms to \( \int_{g(a)}^{g(b)} f(g^{-1}(u)) \cdot \left| \frac{d}{du}(g^{-1}(u)) \right| \, du \). This ensures consistency between the two coordinate systems involved.

Variable Change in Integration

Changing variables, or substitution, is a common technique to make integration easier. By substituting one variable for another, often the function becomes simpler to integrate.

• Begin by identifying a substitution that simplifies the integrand. This might involve trigonometric, exponential, or other functions.
• Derive the differential form of this substitution, adjusting the entire integral accordingly.
• Remember to reflect this substitution in the limits of integration if it's a definite integral.

For example, converting the integral \( \int_0^{\tan(1)/2} \frac{2}{1+u^2} \, du \) by setting \( x = \tan^{-1}(2u) \) transforms the problem entirely. With variable \( x \), the differential \( du \) becomes \( \frac{1}{2} \sec^2(x) \, dx \), significantly simplifying the computation until it becomes \( \int_0^1 \, dx \), which evaluates cleanly to 1. Variable change functions as a powerful tool in the integration arsenal, offering clarity and simplicity to otherwise difficult integrals.