Problem 178
Question
In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{1}^{2} \frac{2}{x^{3}} d x $$
Step-by-Step Solution
Verified Answer
The value of the integral is \( \frac{3}{4} \).
1Step 1: Identify the Problem
We need to evaluate the definite integral \( \int_{1}^{2} \frac{2}{x^{3}} \, dx \) using the Fundamental Theorem of Calculus, Part 2. This theorem states that if \( F \) is an antiderivative of \( f \) on an interval [a, b], then \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \].
2Step 2: Find the Antiderivative
We start by finding an antiderivative of \( f(x) = \frac{2}{x^3} \). Rewrite \( f(x) \) as \( 2x^{-3} \). The antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} \) for \( n eq -1 \). Thus, the antiderivative of \( 2x^{-3} \) is \( F(x) = \frac{2}{-2}x^{-2} = -x^{-2} \).
3Step 3: Evaluate the Antiderivative at the Bounds
Now, apply the Fundamental Theorem of Calculus. We need \( F(2) - F(1) \). Substitute into the antiderivative: \( F(2) = -\frac{1}{2^2} = -\frac{1}{4} \) and \( F(1) = -\frac{1}{1^2} = -1 \).
4Step 4: Calculate the Definite Integral
Replace \( F(b) \) and \( F(a) \) into the formula: \[ \int_{1}^{2} \frac{2}{x^{3}} \, dx = -\frac{1}{4} - (-1) = -\frac{1}{4} + 1 = \frac{3}{4}. \] Thus, the value of the integral is \( \frac{3}{4} \).
Key Concepts
Definite IntegralAntiderivativeIntegration Steps
Definite Integral
A definite integral represents the area under a curve from one point to another on a graph, specifically between the limits or bounds of the integral. In our exercise, the definite integral is given as \( \int_{1}^{2} \frac{2}{x^{3}} \, dx \), with the lower bound being 1, and the upper bound 2.
The definite integral has several important characteristics:
The definite integral has several important characteristics:
- It is calculated over a closed interval [a, b].
- The process involves evaluating the integral function at specific limits.
- The result provides a numerical value representing the net area.
Antiderivative
An antiderivative, also known as an "indefinite integral," is a function that reverses the process of differentiation. In simpler terms, while differentiation breaks down a function to reveal its rate of change, finding an antiderivative compiles the rate of change back into the original function.
When solving the given exercise, the function \( f(x) = \frac{2}{x^3} \) is rewritten as \( 2x^{-3} \). To find its antiderivative, we apply the power rule which states that the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} \) if \( n eq -1 \).
In this case, the antiderivative is determined to be \( F(x) = -x^{-2} \), which simplifies integration by reversing the differentiation process used on the original function.
When solving the given exercise, the function \( f(x) = \frac{2}{x^3} \) is rewritten as \( 2x^{-3} \). To find its antiderivative, we apply the power rule which states that the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} \) if \( n eq -1 \).
In this case, the antiderivative is determined to be \( F(x) = -x^{-2} \), which simplifies integration by reversing the differentiation process used on the original function.
Integration Steps
Integration involves several clear steps when using the Fundamental Theorem of Calculus. These steps help translate the problem into a solvable equation, ensuring an accurate result. Here's a breakdown of the process:
- **Step 1: Identify the Problem:** Recognizing the integral \( \int_{1}^{2} \frac{2}{x^{3}} \, dx \) and the requirement to use the Fundamental Theorem of Calculus.
- **Step 2: Find the Antiderivative:** First, express the integrand in a simpler form if needed (like converting powers). Then, find its antiderivative. Here, it results in \( F(x) = -x^{-2} \).
- **Step 3: Evaluate at the Bounds:** Use the antiderivative to evaluate at the bounds \( F(2) \) and \( F(1) \). This gives the values \( -\frac{1}{4} \) and \( -1 \) respectively.
- **Step 4: Calculate the Result:** Substitute these evaluations into the equation \( F(b) - F(a) \) to find the definite integral value; resulting here in \( \frac{3}{4} \).
These structured steps ensure that we systematically address the problem from finding an antiderivative to calculating the definitive result.
- **Step 1: Identify the Problem:** Recognizing the integral \( \int_{1}^{2} \frac{2}{x^{3}} \, dx \) and the requirement to use the Fundamental Theorem of Calculus.
- **Step 2: Find the Antiderivative:** First, express the integrand in a simpler form if needed (like converting powers). Then, find its antiderivative. Here, it results in \( F(x) = -x^{-2} \).
- **Step 3: Evaluate at the Bounds:** Use the antiderivative to evaluate at the bounds \( F(2) \) and \( F(1) \). This gives the values \( -\frac{1}{4} \) and \( -1 \) respectively.
- **Step 4: Calculate the Result:** Substitute these evaluations into the equation \( F(b) - F(a) \) to find the definite integral value; resulting here in \( \frac{3}{4} \).
These structured steps ensure that we systematically address the problem from finding an antiderivative to calculating the definitive result.
Other exercises in this chapter
Problem 176
In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{4}^{8}\left(4 t^{5 / 2}-3 t^{3 / 2}\rig
View solution Problem 177
In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{1 / 4}^{4}\left(x^{2}-\frac{1}{x^{2}}\r
View solution Problem 179
In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{1}^{4} \frac{1}{2 \sqrt{x}} d x $$
View solution Problem 180
In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{1}^{4} \frac{2-\sqrt{t}}{t^{2}} d t $$
View solution