Problem 10 - Calculus Concepts

Question

For Activities 5 through \(16,\) evaluate the improper integral. $$ \int_{-x}^{-10} 4 x^{-3} d x $$

Step-by-Step Solution

Answer

The value of the improper integral is \(-\frac{2}{x^2} + 0.02\).

Step 1: Understand the Problem

The given problem asks for the evaluation of an improper integral of the function \(4x^{-3}\) from \(-x\) to \(-10\). This integral is improper because the upper limit of the integral, \(-x\), could be a point where the integrand becomes undefined or has infinity.

Step 2: Setup the Integral

The integral is given by:\[\int_{-x}^{-10} 4x^{-3} \, dx\] Here, \(-10\) is the lower limit, and \(-x\) is the upper limit.

Step 3: Find the Antiderivative

To integrate \(4x^{-3}\), we find the antiderivative. The antiderivative of \(x^{-3}\) is \(-\frac{1}{2}x^{-2}\). Thus, the antiderivative of \(4x^{-3}\) is:\[-2x^{-2}\]

Step 4: Evaluate the Definite Integral

Substitute the limits into the antiderivative:\[\left[ -2x^{-2} \right]_{-10}^{-x} = -2(-x)^{-2} - \left( -2(-10)^{-2} \right)\]Calculate each term:\[= -2(-x)^{-2} + 2(0.01) \]\[= -\frac{2}{x^2} + 0.02\]

Step 5: Simplify the Result

The expression for the definite integral becomes:\[-\frac{2}{x^2} + 0.02\]

Step 6: Interpret the Result

With the expression \(-\frac{2}{x^2} + 0.02\), note that the value depends on \(x\). Since this is an improper integral problem, make sure the integral evaluates to a finite number when x approaches values that don't cause division by zero.

Key Concepts

Antiderivative

An antiderivative of a function is like its 'reverse' derivative. If you take a derivative to find how a function is changing, finding an antiderivative tells you about the original function from which it came. The process is often known as integration.When you have a function such as \(4x^{-3}\), finding its antiderivative means looking for a function whose derivative is \(4x^{-3}\). For this problem, the antiderivative of \(x^{-3}\) is \(-\frac{1}{2}x^{-2}\). Therefore, the antiderivative of \(4x^{-3}\) becomes \(-2x^{-2}\).

• Key Idea: Differentiation and antiderivatives are opposites.
• Process: Determine the function that gives you the original upon differentiation.

This is crucial because finding the antiderivative is a step toward calculating the definite integral.

Definite Integral

A definite integral is an integral with upper and lower limits, and it represents the accumulation of quantities, like area under a curve, between these limits. For the given exercise, the definite integral is written as:\[\int_{-x}^{-10} 4x^{-3} \, dx\]With definite integrals, you often substitute the limits into the antiderivative. For instance, substituting the limits into \(-2x^{-2}\), we get:\[\left[ -2x^{-2} \right]_{-10}^{-x} = -2(-x)^{-2} - \left( -2(-10)^{-2} \right)\]

• Importance: Definite integrals provide a specific numerical value.
• Calculation: Substitute and simplify using the antiderivative and the limits.

In this exercise, the integral evaluates to: \(-\frac{2}{x^2} + 0.02\). This expression shows how the value of the integral depends on \(x\), and correct substitution ensures accurate results.

Upper Limit

The upper limit in a definite integral indicates where the integration stops along the x-axis. In this exercise, the upper limit is \(-x\). It's imperative to recognize how changes to the upper limit affect the evaluation of the integral.For the improper integral in the exercise, the function might become undefined if \(x\) ultimately causes a division by zero. This is what differentiates improper integrals from proper ones.

• Role of Upper Limit: Defines the endpoint of integration. Changes in this limit alter the value of the integral.
• Improper Integral Impact: If the upper limit causes the integrand to be undefined (e.g., division by zero), the integral might not converge to a finite value.

Thus, to correctly handle an improper integral, you need to ensure that the limit avoids values that make the expression undefined as \(x\) approaches these critical points. This is part of evaluating whether an improper integral converges or diverges.