91Ó°ÊÓ

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 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.

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.