Problem 20 - Calculus: Early Transcendentals

Question

Set up the integral to compute the arc length of the function on the given interval. Do not evaluate the integral. $$ f(x)=x^{2} \text { on }[0,1] $$

Step-by-Step Solution

Answer

\( L = \int_{0}^{1} \sqrt{1 + 4x^2} \, dx \) is the setup integral for arc length.

Step 1: Recall the Arc Length Formula

The arc length of a function \( y = f(x) \) from \( x = a \) to \( x = b \) is given by the integral formula: \[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]

Step 2: Find the Derivative of the Function

The function given is \( f(x)=x^{2} \). Find the derivative \( \frac{dy}{dx} \) of \( f(x) \):\[ \frac{dy}{dx} = \frac{d}{dx}(x^2) = 2x \]

Step 3: Substitute the Derivative into the Arc Length Formula

Substitute \( \frac{dy}{dx} = 2x \) into the arc length formula:\[ L = \int_{0}^{1} \sqrt{1 + (2x)^2} \, dx \]

Step 4: Simplify the Expression Inside the Integral

Simplify the expression under the square root:\[ L = \int_{0}^{1} \sqrt{1 + 4x^2} \, dx \]

Step 5: Write Down the Final Integral Expression

The integral to compute the arc length of \( f(x) = x^2 \) on \([0, 1]\) is given by:\[ L = \int_{0}^{1} \sqrt{1 + 4x^2} \, dx \]

Key Concepts

Integral Calculus

Integral calculus is a fundamental branch of calculus that focuses on accumulation and area under curves. At its core, it deals with integrals, which can be thought of as the reverse process of differentiation. Instead of finding the rate of change, integral calculus answers questions about accumulation, such as finding areas, volumes, and, in this exercise, the arc length of a curve.

In practical applications, integral calculus helps us compute quantities when geometric or algebraic forms aren't easily expressible. In this exercise, we are tasked to set up an integral to compute the arc length of the function \( f(x) = x^2 \) on the interval \([0, 1]\). The arc length integral uses the formula \( L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \). This formula builds on the idea of summing up small increments of length along the curve, which is an abstraction of adding small straight line segments to approximate the curve's length.

Understanding integral calculus through such applications allows us to tackle and solve real-world problems involving curves and shapes.

Derivative

A derivative is a central concept in calculus that denotes the rate at which a function is changing at any given point. It's essentially the slope of the function's graph at that point. For any function \( f(x) \), the derivative \( \frac{dy}{dx} \) measures how \( y \) changes with respect to \( x \). The process of finding a derivative is known as differentiation.

In our exercise, deriving the function \( f(x) = x^2 \) results in \( \frac{dy}{dx} = 2x \). This derivative is used in the arc length formula to determine how the curve’s direction and steepness vary along the given interval \([0, 1]\). It's essential because the arc length formula requires the square of this derivative within the square root to provide a measure of the function's curve.

Understanding derivatives helps analyze and predict the behavior of functions, making it crucial in both mathematical theory and practical problem-solving.

Arc Length Formula

The arc length formula is a crucial tool in integral calculus, used to measure the distance along a curve between two points. It allows us to find the length of curves that don’t follow a straight line. The formula is: \[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \] Here, \( \frac{dy}{dx} \) is the derivative of the curve \( y = f(x) \), and \( [a, b] \) is the interval along which you are calculating the curve's length.

In our context, applying this formula involves substituting the derived \( \frac{dy}{dx} = 2x \) into it, resulting in \( L = \int_{0}^{1} \sqrt{1 + 4x^2} \, dx \). This integral expression captures the geometric intricacies of the parabola between \( x = 0 \) and \( x = 1 \).

By setting up this integral expression, you lay the foundation for calculating the arc length without evaluating it directly. This setup is essential in many mathematical and engineering contexts where understanding the curvature and physical length of an object is necessary.