Problem 31 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-34\) without using tables. $$ \int_{-1}^{4} \frac{d x}{\sqrt{|x|}} $$

Step-by-Step Solution

Answer

The integral evaluates to 6.

Step 1: Understand the Function

The integral given is \( \int_{-1}^{4} \frac{d x}{\sqrt{|x|}} \). We have to be careful with the absolute value in the denominator since it affects the integration process depending on whether \( x \) is positive or negative.

Step 2: Split the Integral

Because of the absolute value, the integral has to be split into two parts where the expression inside the absolute value is positive and negative, respectively. Therefore, the integral will be split from \( -1 \) to \( 0 \) and from \( 0 \) to \( 4 \), given by: \[ \int_{-1}^{0} \frac{d x}{\sqrt{|x|}} + \int_{0}^{4} \frac{d x}{\sqrt{|x|}}. \]

Step 3: Simplify Each Portion

In the first portion \( \int_{-1}^{0} \frac{d x}{\sqrt{|x|}} \), \( |x| = -x \). It simplifies to \( \int_{-1}^{0} \frac{d x}{\sqrt{-x}} \). For the second integral \( \int_{0}^{4} \frac{d x}{\sqrt{|x|}} \), \( |x| = x \), so it remains \( \int_{0}^{4} \frac{d x}{\sqrt{x}} \).

Step 4: Evaluate Each Integral

For \( \int_{-1}^{0} \frac{d x}{\sqrt{-x}} \), make a substitution \( u = -x \), which implies \( du = -dx \). Therefore, the integral becomes \( \int_{1}^{0} \frac{-du}{\sqrt{u}} \), which is \( \int_{0}^{1} \frac{du}{\sqrt{u}} \). Integrate to get \( 2 u^{1/2} \Big|_{0}^{1} = 2(1-0) = 2 \).For \( \int_{0}^{4} \frac{d x}{\sqrt{x}} \), direct integration gives \( 2x^{1/2} \Big|_{0}^{4} = 4 - 0 = 4 \).

Step 5: Combine the Results

Add the results of the two integrals: \( 2 + 4 = 6 \). Therefore, \( \int_{-1}^{4} \frac{d x}{\sqrt{|x|}} = 6 \).

Key Concepts

Absolute Value

Absolute value is a concept that takes any real number and outputs its non-negative portion. For the expression \(|x|\), if \(x\) is positive or zero, it simply equals \(x\) itself. However, if \(x\) is negative, \(|x|\) becomes \(-x\). Understanding how absolute value impacts mathematical expressions is crucial, especially when integrating. In calculus, we often encounter expressions like \(|x|\) that require special consideration when calculating things like integrals.

• This affects the computation since the integral must be adjusted accordingly.
• The behavior of the function changes depending on whether \(x\) is above or below zero.

In integration involving \(|x|\), like in the original exercise, it is necessary to split the integral at the point where \(x\) changes sign, which is typically zero. That allows for handling separate scenarios for when \(x\) is greater or less than zero.

Definite Integral

A definite integral is a form of integration that gives a numerical value representing the area under a curve over a specified interval. In the expression \(\int_{a}^{b} f(x)\, dx\), \(a\) and \(b\) are the lower and upper limits, restricting the area calculation to this defined region. Definite integrals have exciting properties:

• They involve evaluating the integral of a function \(f(x)\) over a specific interval \([a, b]\).
• They directly calculate the net area, considering parts below the x-axis as negative contribution.

In the original exercise, the integral is from \(-1\) to \(4\). Definite integrals give us a precise area value, and since the exercise involves an absolute value function, it implies we need careful evaluation in axial sections.

Integration Techniques

Integration techniques help solve integrals that aren't straightforward. They provide strategies to transform complex problems into solvable forms. In many cases, especially with functions involving absolute values or transformations, these techniques become essential:

• Substitution: A method where we select a new variable to simplify the integral's form. It is particularly useful if the integral involves products or composite functions.
• Splitting the Integral: This technique involves breaking down an integral into two or more parts. Useful when dealing with piecewise functions or discontinuities, like absolute values, as seen in the original exercise.
• Direct Integration: Used when the function fits standard integral tables, though tables weren't used here, being straightforward.

The original exercise utilized these techniques by splitting the integral at the discontinuity point of \(x = 0\), then evaluating separately using substitution where needed.