Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-14, use the Second Fundamental Theorem of Calculus to evaluate each definite integral. \(\int_{1}^{3} \frac{2}{t^{3}} d t\)

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{16}{9} \).

Step 1: Identify the integral type

The given integral is \( \int_{1}^{3} \frac{2}{t^3} \, dt \). It is a definite integral that needs to be evaluated using the Second Fundamental Theorem of Calculus.

Step 2: Apply the Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus states that if \( F \) is an antiderivative of \( f \) on an interval \( [a, b] \), then \( \int_{a}^{b} f(t) \, dt = F(b) - F(a) \). Here, we need to find the antiderivative of \( \frac{2}{t^3} \).

Step 3: Find the antiderivative

The function \( \frac{2}{t^3} \) can be rewritten as \( 2t^{-3} \). To find its antiderivative, integrate: \( \int 2t^{-3} \, dt \). Using the power rule, the antiderivative is \( F(t) = -t^{-2} = -\frac{1}{t^2} \cdot 2 + C \), where \( C \) is a constant. Thus, \( F(t) = -\frac{2}{t^2} \).

Step 4: Evaluate the definite integral using the antiderivative

Using \( F(t) = -\frac{2}{t^2} \), evaluate the definite integral: \( F(3) - F(1) \). First, find \( F(3) = -\frac{2}{3^2} = -\frac{2}{9} \). Then find \( F(1) = -\frac{2}{1^2} = -2 \).

Step 5: Calculate the result

Substitute the values from Step 4: \( F(3) - F(1) = -\frac{2}{9} - (-2) \). This simplifies to \( -\frac{2}{9} + 2 = \frac{16}{9} \).

Key Concepts

definite integrals

A definite integral is a way to find the area under a curve within a specified interval. In mathematics, this is expressed as \( \int_{a}^{b} f(t) \, dt \), where \( a \) and \( b \) are the limits of integration, and \( f(t) \) is the function being integrated. The result of a definite integral is a number, which in essence, is the net area between the function \( f(t) \) and the x-axis from \( a \) to \( b \).

Definite integrals can be used to calculate things like total distance traveled given a velocity-time graph or total accumulated change. They are a crucial tool in calculus, connecting the concept of accumulation and change. In our example, we are working with the integral \( \int_{1}^{3} \frac{2}{t^{3}} \, dt \). Here, we find the area under \( \frac{2}{t^3} \) from \( t = 1 \) to \( t = 3 \). This helps to evaluate physical concepts like changes in speed or cp concentration over a time interval.

antiderivative

An antiderivative is a function whose derivative equals the original function. If \( F(t) \) is an antiderivative of \( f(t) \), then the derivative of \( F(t) \) is \( f(t) \). This process is often called integration, and it is essentially reversing differentiation.

Finding the antiderivative is necessary when evaluating definite integrals. The Second Fundamental Theorem of Calculus exploits this idea by allowing us to compute a definite integral if we know an antiderivative of the function. In the exercise, we need to find an antiderivative of \( \frac{2}{t^3} \). The function can be rewritten as \( 2t^{-3} \) to use standard rules for integration, leading us to the antiderivative \( -\frac{2}{t^2} \), up to a constant \( C \). This shows that differentiation and integration are inverse processes in mathematics.

power rule

The power rule is a fundamental technique in calculus for finding derivatives and integrals of expressions involving exponents. For differentiation, if \( n \) is any real number, then the derivative of \( t^n \) is \( nt^{n-1} \).

For antiderivatives, the rule states that the integral of \( t^n \) is \( \frac{t^{n+1}}{n+1} + C \), provided \( n eq -1 \). This rule gives an efficient method to find antiderivatives of polynomial functions.

In the given problem, apply the power rule to \( 2t^{-3} \). The power rule lets us integrate it to find its antiderivative:

1. Add one to the exponent: \( -3 + 1 = -2 \).
2. Divide the term by the new exponent: \( 2t^{-2}/-2 \), simplifying to \( -\frac{2}{t^2} \).
3. Don't forget the constant, \( C \), but it drops out in definite integrals.

The power rule simplifies the task of finding antiderivatives in coursework, especially for polynomial functions.

calculus step-by-step solution

Breaking down a problem using steps simplifies complex calculus problems. Especially when using the Second Fundamental Theorem of Calculus, a clear sequence of steps ensures correct evaluation of the integral. Here’s a detailed approach:

• Identify the integral: Start by recognizing what type of integral you are solving. For \( \int_{1}^{3} \frac{2}{t^3} \, dt \), it's a definite integral.
• Find the antiderivative: Rewriting \( \frac{2}{t^3} \) as \( 2t^{-3} \), use the power rule to determine the antiderivative as \( -\frac{2}{t^2} \).
• Apply the theorem: The Second Fundamental Theorem of Calculus links antiderivatives with definite integrals. Compute \( F(b) - F(a) \), where \( F(t) \) is the antiderivative.
• Calculate values: Substitute \( a \) and \( b \) (1 and 3 in this case) into \( F(t) \) to find \( F(3) \) and \( F(1) \). Then, compute \( F(3) - F(1) \) for the solution.
• Simplify the result: Calculate the numeric evaluation to derive the final value, \( \frac{16}{9} \).

This methodical approach ensures accuracy and helps build a solid foundation for processing calculus problems efficiently.