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Magazine Chapter 5: Problem 44
What is wrong with the equation? $$\int_{-1}^{2} \frac{4}{x^{3}} d x=-\frac{2}{x^{2}} ]_{-1}^{2}=\frac{3}{2}$$
Short Answer
Expert verified
The original integral is improper since the function is undefined at \(x = 0\).
Step by step solution
01
Identify the Problem
The equation is evaluating the integral of the function \( \frac{4}{x^3} \) from \(x = -1\) to \(x = 2\) using an antiderivative \(-\frac{2}{x^2}\). We need to check if this antiderivative is correct.
02
Differentiation Check
To verify if \(-\frac{2}{x^2}\) is the correct antiderivative, we differentiate it with respect to \(x\). The derivative of \(-\frac{2}{x^2}\) is \( \frac{4}{x^3}\), which confirms that the antiderivative was correctly found.
03
Evaluate the Antiderivative at Limits
Use the Fundamental Theorem of Calculus on the interval \([-1, 2]\) with the antiderivative to find:\[-\frac{2}{2^2} - \left(-\frac{2}{(-1)^2}\right) = -\frac{2}{4} + \frac{2}{1} = -\frac{1}{2} + 2 \]
04
Simplify the Result
Simplify the evaluated expression: \(-\frac{1}{2} + 2 = \frac{3}{2}\).
05
Re-evaluate the Limits of Integration
Although the calculation appears correct, pay attention that the function \(\frac{4}{x^3}\) is not defined at \(x=0\), belonging to the integration interval. This makes the original definite integral improper.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Antiderivative
An antiderivative is a function whose derivative produces the original function you started with. In the context of improper integrals, identifying the correct antiderivative is crucial. For example, if you have a function \( f(x) = \frac{4}{x^3} \), an antiderivative could be \( F(x) = -\frac{2}{x^2} \). To check if you've found the right antiderivative, differentiate it, and see if you arrive back at the original function.
- Start with the original function: \( \frac{4}{x^3} \).
- Propose an antiderivative: \( -\frac{2}{x^2} \).
- Differentiation of \( -\frac{2}{x^2} \) should yield \( \frac{4}{x^3} \).
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration, providing a way to evaluate definite integrals easily. It states that if \( F \) is an antiderivative of \( f \) on an interval \([a, b]\), then:\[\int_{a}^{b} f(x) \, dx = F(b) - F(a)\]This theorem allows us to calculate the area under a curve defined by \( f(x) \) between two points \( a \) and \( b \) by simply evaluating the antiderivative \( F \) at these points and subtracting.
- Calculate \( F(b) \), the antiderivative at the upper limit \( b \).
- Calculate \( F(a) \), the antiderivative at the lower limit \( a \).
- The integral's value is \( F(b) - F(a) \).
Differentiation
Differentiation is the process of finding a derivative. It's the reverse of finding an antiderivative. In cases involving improper integrals, differentiation is used to verify that a proposed antiderivative is correct. We handle this by working through the derivative step-by-step:Given an antiderivative \( F(x) = -\frac{2}{x^2} \), use differentiation rules to find the original function. For instance, for \(-\frac{2}{x^2} \), compute as follows:- Apply the power rule: \( (x^n)' = n \cdot x^{n-1} \).- Here, \( x^n = x^{-2} \), so \( F'(x) = -2 \cdot x^{-3} = \frac{4}{x^3} \).Successful differentiation confirms the accuracy of the antiderivative.
Limits of Integration
The limits of integration define the range over which we calculate the area under a curve when evaluating a definite integral. They are crucial in the context of improper integrals, as these involve ranges where the function might not be well-defined or infinite.
- In the integral \( \int_{-1}^{2} \) of \( \frac{4}{x^3} \, dx \), the limits of integration are from \(-1\) to \(2\).
- This covers an interval containing \(x = 0\), where the function is undefined because it involves division by zero.
- This makes the integral improper, requiring special considerations, such as splitting the integral or using limits to handle the undefined part.
