Problem 5 - Calculus

Question

Use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{1}^{4} \frac{1}{w^{2}} d w $$

Step-by-Step Solution

Answer

The definite integral evaluates to \( \frac{3}{4} \).

Step 1: Identify the Function to Integrate

We need to evaluate the integral \( \int_{1}^{4} \frac{1}{w^2} \, dw \). This involves finding an antiderivative for the integrand \( \frac{1}{w^2} \) first.

Step 2: Rewrite the Integrand

The integrand \( \frac{1}{w^2} \) can be rewritten as \( w^{-2} \). This makes it easier to apply the power rule for integration.

Step 3: Find the Antiderivative

Using the power rule which states \( \int w^n \, dw = \frac{w^{n+1}}{n+1} + C \), we find the antiderivative of \( w^{-2} \). Thus, \( \int w^{-2} \, dw = \frac{w^{-1}}{-1} + C = -w^{-1} + C \).

Step 4: Evaluate the Definite Integral

Apply the Second Fundamental Theorem of Calculus, which tells us to evaluate the antiderivative from 1 to 4: \( \left[ -w^{-1} \right]_{1}^{4} = -\left( \frac{1}{4} \right) - \left( -\frac{1}{1} \right) \).

Step 5: Simplify the Expression

Simplify the expression from Step 4: \(-\frac{1}{4} + 1 = 1 - \frac{1}{4} = \frac{3}{4} \).

Key Concepts

Antiderivative

An antiderivative is a function that reverses the process of differentiation. If you think of differentiation as finding the rate at which something changes, then finding an antiderivative is like discovering the original function from that rate.
For example, if you differentiate a function and get \( \frac{1}{w^2} \), then the antiderivative will tell you what function, when differentiated, results in \( \frac{1}{w^2} \). This is crucial for solving integrals, especially in calculus where you'll often encounter functions you need to integrate.
In the given exercise, we start by finding the antiderivative of \( \frac{1}{w^2} \). By rewriting \( \frac{1}{w^2} \) as \( w^{-2} \), it becomes easier to integrate using the power rule. The antiderivative we end up with is \( -w^{-1} \), also expressed as \(-\frac{1}{w}\).
Remember, any antiderivative includes a constant of integration, \( C \), because differentiation of a constant (") contributes nothing to the derivative.

Definite Integral

A definite integral calculates the net area between the function's curve and the x-axis, over a specific interval. It provides a numerical value rather than another function. This process is fundamental in calculus as it deals with accumulation.
In the exercise, the definite integral \( \int_{1}^{4} \frac{1}{w^2} \, dw \) requires evaluating the antiderivative we found—\(-w^{-1}\)—between the limits of 1 and 4. When we apply the Second Fundamental Theorem of Calculus, it states that the definite integral of a function over an interval is the difference in the values of its antiderivative evaluated at the endpoints of that interval.
For our integral, this means calculating \([-\frac{1}{w}]_{1}^{4}\), which gives us \(-\frac{1}{4} - (-1)\). This results in \(\frac{3}{4}\), representing the net area from 1 to 4.

• Definite integrals provide the total amount or net change across an interval.
• They require precise evaluation of the antiderivative at given limits.

Power Rule for Integration

The power rule for integration is a straightforward technique to find integrals, especially for polynomial functions. It simplifies the process by providing a formula. If you have an integrand in the form \( w^n \), the power rule states that the antiderivative is \( \frac{w^{n+1}}{n+1} + C \), as long as \( n eq -1 \).
This rule is applicable in the exercise, where the integrand \( \frac{1}{w^2} = w^{-2} \) is present. By using the power rule, we adjust the exponent by adding one, resulting in \(-w^{-1}\) as our antiderivative.

• It simplifies finding antiderivatives, making it a handy tool in calculus.
• Its limitation is when \(n = -1\), requiring a different approach, such as logarithms.

In conclusion, the power rule aids in transforming a polynomial-style integrand into its antiderivative efficiently, which then is either used to describe accumulated changes or evaluate definite integrals as in this problem.