Problem 24 - Applied Calculus

Question

Find an antiderivative. $$p(r)=2 \pi r$$

Step-by-Step Solution

Answer

The antiderivative is \(\pi r^2 + C\).

Step 1: Identify the Function

We are given the function \(p(r) = 2 \pi r\). Our task is to find the antiderivative, which involves integrating the function with respect to \(r\).

Step 2: Apply the Power Rule for Integration

To find the antiderivative of \(p(r) = 2 \pi r\), we can use the power rule for integration. The power rule states that \(\int r^n \, dr = \frac{r^{n+1}}{n+1} + C\) for \(n eq -1\). In this case, \(r^1\) applies, so we set \(n = 1\).

Step 3: Calculate the Antiderivative

Integrating \(2 \pi r\) with respect to \(r\), we apply the power rule: \[ \int 2 \pi r \, dr = 2 \pi \int r \, dr = 2 \pi \left( \frac{r^{1+1}}{1+1} \right) + C = 2 \pi \left( \frac{r^2}{2} \right) + C \].

Step 4: Simplify the Expression

Simplifying the expression gives: \[ 2 \pi \left( \frac{r^2}{2} \right) + C = \pi r^2 + C \]. The constant \(C\) represents the constant of integration.

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental technique used to find antiderivatives, especially when dealing with polynomial functions. It states that if you want to integrate a function of the form \( r^n \), where \( n \) is any real number except -1, you can apply the formula:

• \( \int r^n \, dr = \frac{r^{n+1}}{n+1} + C \)

Here, \( C \) represents the constant of integration, which we'll talk about later.
This rule simplifies the process of finding antiderivatives by increasing the power of \( r \) by one and then dividing by the new power.
Imagine you have \( 2 \pi r \) as in our exercise above. Essentially, \( r \) is \( r^1 \) so by the power rule:

• You increase the exponent 1 to 2, giving you \( r^2 \).
• Then, you divide by the new exponent, 2, resulting in \( \frac{r^2}{2} \).

Thus, integrating \( 2 \pi r \) results in \( 2 \pi \left( \frac{r^2}{2} \right) \).
This kind of rule helps you work through integration without needing to guess or check multiple different methods.

Integration

Integration is the process of finding an antiderivative of a function. It is the reverse operation of differentiation.
When you differentiate a function, you find its rate of change. Integration finds the original function given that rate of change.
In the context of our exercise, you're given \( p(r) = 2 \pi r \) and you are tasked with determining the function whose derivative is \( 2 \pi r \).

• This process involves applying rules like the power rule, which makes finding the antiderivative simpler and more systematic.
• Importantly, unlike differentiation, integration can introduce a constant of integration, because many functions can have the same derivative except for a constant difference.
• This means for a function \( f(r) \), multiple antiderivatives are possible that look like \( F(r) = f(r) + C \).

So, when integrating, always remember to include this constant to account for all possible original functions.

Constant of Integration

When we integrate a function, we often add a term known as the constant of integration, denoted as \( C \).
Why is this necessary? Let's explore:

• Consider that differentiation eliminates constants; for example, both \( r^2 + 3 \) and \( r^2 - 4 \) simplify to a derivative of \( 2r \).
• This means during integration, you cannot decipher specific constants just from taking the antiderivative unless given more information.
• Thus, integration must include \( C \) — it incorporates an indefinite number of vertical shifts of the antiderivative function that still fit the derivative presented.

In other words, integrating \( 2 \pi r \) results in \( \pi r^2 + C \).
The \( C \) here embodies all possible unknown constants for which the derivative is \( 2 \pi r \).
It's a crucial step that guarantees the completeness of our solution when finding antiderivatives.