Question

Set up the integral to compute the arc length of the function on the given interval. Do not evaluate the integral. \(f(x)=x^{10}\) on [0,1]

Short answer

\( \int_{0}^{1} \sqrt{1 + 100x^{18}} \, dx \)

Step-by-step solution

Identify the Arc Length Formula

The arc length of a function from \( x = a \) to \( x = b \) is given by the integral \( \int_{a}^{b} \sqrt{1 + \left( f'(x) \right)^2} \, dx \). For this problem, \( f(x) = x^{10} \) and the interval is \([0, 1]\).

Find the Derivative of the Function

First, find the derivative of \( f(x) = x^{10} \). Using the power rule, we have \( f'(x) = 10x^9 \).

Set Up the Integrand

Substitute \( f'(x) = 10x^9 \) into the arc length formula's integrand: \( \sqrt{1 + (10x^9)^2} = \sqrt{1 + 100x^{18}} \).

Set Up the Definite Integral

Substitute the integrand into the integral for the arc length over the interval \([0,1]\): \( \int_{0}^{1} \sqrt{1 + 100x^{18}} \, dx \). This is the integral that represents the arc length, though we do not need to evaluate it for this exercise.

Key concepts

Integral Setup

When we say "integral setup," we're talking about how we structure an integral to solve a specific problem. In calculus, integrals help us find many things, including the area under a curve or the arc length of a function. To set up an integral for arc length,we need a formula that captures how the length changes over a specific interval. In this exercise, we start with the function given, which is \(f(x) = x^{10}\), and the interval \([0, 1]\).

The aim is to utilize the arc length formula, which will help us in defining the structure of our integral. By following the proper setup, we'll accurately represent the desired arc length of any smooth curve over the interval.

Derivative

The word "derivative" comes from calculus and refers to the rate at which a function changes at any point. In layman's terms, it's like the slope of the curve of the function at a given point. In our context, we first need to find the derivative of the function \(f(x) = x^{10}\). This step is key because the derivative is part of the integrand in the arc length formula.

To find the derivative of \(x^{10}\), we use the power rule, which provides a simple way to differentiate terms of the form \(x^n\). Here, \(n = 10\), so applying the power rule gives us \(f'(x) = 10x^9\). Knowing this helps us to move forward in setting up the integrand.

Arc Length Formula

The arc length formula is a cornerstone in calculus when we need to find the length of a curve between two points.For any continuous function, the arc length from \(x = a\) to \(x = b\) can be determined using the formula:\[\int_{a}^{b} \sqrt{1 + \left( f'(x) \right)^2} \, dx\]This formula essentially integrates the infinitesimally small lengths along the curve to get the total arc length.

In our exercise, with \(f'(x) = 10x^9\), we substitute this into our arc length formula's integrand:\(\sqrt{1 + (10x^9)^2} = \sqrt{1 + 100x^{18}}\). This transformed expression is part of the integrand whose total integral over [0, 1] represents the length of the curve.

Power Rule

The power rule is a helpful tool in calculus for finding the derivative of a function. When you have a function in the form of \(x^n\), the power rule lets you quickly determine its derivative: \(nx^{n-1}\).

In our step-by-step solution, to find the derivative of \(f(x) = x^{10}\), we applied the power rule. By doing so, we found that \(f'(x) = 10x^9\). The power rule simplifies the process of differentiation, especially when dealing with polynomial functions. Understanding how to apply this rule is crucial for setting up integrals for problems involving arc length, optimizing how we compute derivatives for each term in our function.