Problem 11 - Calculus Early Transcendentals

Question

Arc length by calculator a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=x^{2} ;[-1,1]$$

Step-by-Step Solution

Answer

Answer: The approximate arc length of the curve $$y = x^2$$ on the interval $$[-1,1]$$ is 2.433.

Step 1: Find the derivative of the function y(x)

The given function is $$y(x) = x^2$$. Let's find its derivative with respect to x, which we'll denote as $$y'(x)$$: $$y'(x) = \frac{d}{dx} (x^2) = 2x$$

Step 2: Write the arc length formula and substitute in the function and derivative

Now that we have the derivate of the function, we can use the arc length formula, $$L = \int_{a}^{b} \sqrt{1 + (y'(x))^2} \, dx$$. Substitute the function and its derivative into the formula: $$L = \int_{-1}^{1} \sqrt{1 + (2x)^2} \, dx$$

Step 3: Simplify the integral

Now simplifying the integral: $$L = \int_{-1}^{1} \sqrt{1 + 4x^2} \, dx$$ At this point, this integral cannot be easily analytically solved, so we need to use technology (e.g., a calculator or computer software) to approximate the integral.

Step 4: Approximate the integral using technology

Using a calculator or software like Wolfram Alpha, we can input the integral to get the approximate arc length: $$L \approx 2.433$$ Thus, the arc length of the curve $$y = x^2$$ on the interval $$[-1,1]$$ is approximately 2.433.

Key Concepts

Integral Evaluation

When calculating the arc length of a curve, one key step is evaluating the integral that represents this length. The integral is defined using the arc length formula:

\[ L = \int_{a}^{b} \sqrt{1 + (y'(x))^2} \, dx \] Here, \(a\) and \(b\) are the limits of the interval, and \(y'(x)\) is the derivative of the function. Integral evaluation involves calculating this integral to find the arc length. In our specific problem, the integral is:

\[ L = \int_{-1}^{1} \sqrt{1 + 4x^2} \, dx \] This integral is not straightforward to solve analytically because of the square root and the quadratic expression inside it. Therefore, when such integrals are complex, we often rely on computational tools or calculators to evaluate or approximate the integral. These technologies use numerical methods to provide approximate values. In this case, the computational approach gives an approximate arc length of around 2.433.

Derivative Calculation

Understanding how to find and use the derivative is crucial in determining arc length. Let's dive into the process. The derivative represents the rate of change of a function concerning its variable. For the function \( y(x) = x^2 \), we calculate the derivative with respect to \( x \) to get \( y'(x) \).

**Derivative Calculation Process:**

• The derivative of \( x^2 \) with respect to \( x \) is determined by using the power rule: \( \frac{d}{dx}(x^n) = nx^{n-1} \).
• Apply the power rule: \( \frac{d}{dx}(x^2) = 2 \cdot x^{2-1} = 2x \).

This derivative \( y'(x) = 2x \) plays a critical role in forming the integrand for the arc length formula. Specifically, it appears as \( (y'(x))^2 \) or \((2x)^2\) when substituted into the formula. Hence, mastering derivative calculation ensures that you can accurately find and work with integrals in arc length problems.

Square Root Simplification

Square root simplification is often a significant part of solving problems involving arc length. Once the derivative has been found and squared, it is added to 1 underneath a square root as part of the arc length integrand:

\[ \sqrt{1 + (y'(x))^2} \].

For our function, the expression to simplify becomes:

\[ \sqrt{1 + (2x)^2} = \sqrt{1 + 4x^2} \].

Simplifying expressions under the square root is crucial because it can sometimes simplify the integral calculation process or make numerical evaluation easier. However, in some cases, like ours, even after simplification, the integral remains complex.
At this point, further simplification may not help. We then resort to computational methods to evaluate the complexity analytically, giving an approximate value using tools like a calculator, and in this specific problem, approximating to 2.433. Always remember, simplification aims to make evaluation more feasible, either manually or with technology.