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Chapter 3: Problem 4

In this test: Unless otherwise specified, the domain of a function \(f\) is assumed to be the set of all real numbers \(x\) for which \(f(x)\) is a real number. \(\int_{1}^{3} \frac{8}{x^{3}} d x=\) (A) \(\frac{32}{9}\) (B) \(\frac{40}{9}\) (C) 0 (D) \(-\frac{32}{9}\)

Short Answer

Expert verified
The correct answer is (A) \(\frac{32}{9}\).

Step by step solution

01

Integrate the Function

First, you need to find the anti-derivative (indefinite integral) of the function. The integral of \(\frac{8}{x^{3}} d x\) is \(-\frac{4}{x^2}\).
02

Apply Fundamental Theorem of Calculus

The next step is to apply the Fundamental Theorem of Calculus, which states that the definite integral of a function can be found by subtracting the value of its anti-derivative evaluated at the lower limit of the integral from its value evaluated at the upper limit. In this case, you subtract the value of \(-\frac{4}{x^2}\) evaluated at x=1 from its value evaluated at x=3.
03

Substitute the Limits

Substitute the limits 1 and 3 into \(-\frac{4}{x^2}\). You'll get at x=3 is \(-\frac{4}{3^2}\) which is \(-\frac{4}{9}\) and at x=1 is \(-\frac{4}{1^2}\) which equals -4.
04

Perform the Subtraction

Subtract the two results. This gives you \(-\frac{4}{9} - (-4) = -\frac{4}{9} + 4 = \frac{32}{9}\)
05

Match with the Options

Finally, match the result \(\frac{32}{9}\) with the options (A)-(D), whereby it matches with (A).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is a bridge between differentiation and integration. It elegantly states that if you have the anti-derivative of a function, you can find the definite integral over a specific interval. This theorem has two main parts:
  • The first part tells us that the integration of a function can be reversed by differentiation.
  • The second part allows us to use anti-derivatives to evaluate definite integrals. It says that the integral from $a$ to $b$ of $f(x)$ is given by $F(b) - F(a)$, where $F(x)$ is an anti-derivative of $f(x)$.
Understanding this allows us to quickly compute definite integrals without directly calculating the area under a curve as an infinite series.
Anti-derivatives
Anti-derivatives are the reverse operation of derivatives. When finding an anti-derivative, we're essentially asking: "What function would give us this derivative?" In the context of the exercise, the function given is \(\frac{8}{x^3}\), and its anti-derivative is calculated to be \(-\frac{4}{x^2}\). This tells us that if we differentiate \(-\frac{4}{x^2}\), we will obtain \(\frac{8}{x^3}\).
This process is crucial as it sets the foundation for using the Fundamental Theorem of Calculus. Without finding the anti-derivative, evaluating a definite integral becomes a lot more complex.
Indefinite Integrals
Indefinite integrals refer to finding the anti-derivative of a function, but unlike definite integrals, they don鈥檛 evaluate over a specific interval. They generally have the form \( \int f(x)\, dx = F(x) + C \), where \(C\) is a constant of integration.
The purpose of indefinite integrals is to represent a family of functions that could all be potential anti-derivatives. In our exercise, the indefinite integral \(-\frac{4}{x^2}\) is crucial for finding the definite value later using the Fundamental Theorem of Calculus. It's like preparing a map before planning a journey.
Integration Techniques
Integration techniques are strategies to solve integrals, especially when they're not straightforward. Some common techniques include substitution, integration by parts, and partial fractions.
In the given exercise, a straightforward power rule was applied to find the anti-derivative. Whenever you have a function like \(\frac{8}{x^3}\), it can be rewritten as \(8x^{-3}\), making it easy to apply the basic power rule of integration, where \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).
Learning different techniques helps tackle more complex integrals confidently and efficiently.

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Most popular questions from this chapter

In this test: Unless otherwise specified, the domain of a function \(f\) is assumed to be the set of all real numbers \(x\) for which \(f(x)\) is a real number. The value of \(c\) that satisfies the Mean Value Theorem for derivatives on the interval \([0,5]\) for the function \(f(x)=x^{3}-6 x\) is (A) 0 (B) 1 (C) \(\frac{5}{3}\) (D) \(\frac{5}{\sqrt{3}}\)

In this test: Unless otherwise specified, the domain of a function \(f\) is assumed to be the set of all real numbers \(x\) for which \(f(x)\) is a real number. If \(3 x^{2}-2 x y+3 y=1,\) then when \(x=2, \frac{d y}{d x}=\) (A) \(-12\) (B) \(-10\) (C) \(-\frac{10}{7}\) (D) \(12\)

In this test: Unless otherwise specified, the domain of a function \(f\) is assumed to be the set of all real numbers \(x\) for which \(f(x)\) is a real number. If \(f(x)=\int_{0}^{x+1} \sqrt[3]{t^{2}-1},\) then \(f^{\prime}(-4)=\) (A) \(-2\) (B) 2 (C) \(\sqrt[3]{15}\) (D) 0
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