Problem 13 - Thomas Calculus

Question

In Exercises \(13-20,\) do the following. a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher's or computer's integral evaluator to find the curve's length numerically. $$y=x^{2},-1 \leq x \leq 2$$

Step-by-Step Solution

Answer

The curve length is approximately 5.3884.

Step 1: Set Up the Integral for Curve Length

The length of the curve for the function \(y = x^2\) from \(x = -1\) to \(x = 2\) can be found using the formula for arc length in calculus: \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \] where \( \frac{dy}{dx} \) is the derivative of \(y\) with respect to \(x\). For \(y = x^2\), \( \frac{dy}{dx} = 2x \). Substitute this into the formula to get: \[ L = \int_{-1}^{2} \sqrt{1 + (2x)^2} \, dx = \int_{-1}^{2} \sqrt{1 + 4x^2} \, dx \] This is the integral that represents the length of the curve.

Step 2: Graph the Curve

Graph the function \(y = x^2\) over the interval \(-1 \leq x \leq 2\) to visualize the curve. This is a parabolic curve opening upwards. A graph will help in understanding the path whose length we are calculating. A simple sketch or using graphing software can be used to plot the curve.

Step 3: Evaluate Integral Numerically

To find the numerical value of the integral representing the curve's length, use a graphing calculator or computer software. Input the integral \(\int_{-1}^{2} \sqrt{1 + 4x^2} \, dx \). Most calculators will provide a numeric approximation directly. The evaluated length should be approximately \(5.3884\) after solving it numerically.

Key Concepts

Integral Calculus

Integral calculus is a cornerstone of mathematics that deals mainly with integrals and their applications. Integrals help us find quantities like area, volume, and in our exercise, the arc length of a curve.

To calculate the arc length of a curve, we use a specific integral formula. The formula looks like this:

• Length of curve: defined from the integral \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]

In our case, we applied this formula to the function \(y = x^2\) over the interval \(-1 \leq x \leq 2\). This is why we need to calculate this integral \[ \int_{-1}^{2} \sqrt{1 + 4x^2} \, dx \] in order to find the arc length. Understanding how to set up and compute such integrals is key to solving problems in calculus that involve determining lengths or other cumulative measures.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. It's like finding the slope of a curve at any particular point.

In the context of our exercise, we calculate the derivative of \(y = x^2\) to work on the arc length formula.

• The derivative \( \frac{dy}{dx} \) of the function \(y = x^2\) is determined by differentiating it with respect to \(x\), which is \(2x\).

This derivative is then squared and added to 1, forming part of the integrand in the formula for arc length:
\[ 1 + \left(2x\right)^2 = 1 + 4x^2 \]

Knowing how to calculate and apply derivatives is essential for both understanding changes in functions and solving problems involving rates of change and tangencies.

Numerical Integration

Not all integrals can be easily solved analytically. Sometimes, especially in real-world applications, we use numerical integration.

Numerical integration helps us approximate the integral's value using algorithms. These methods are embedded in calculators and software, making them practical tools.

• In this exercise, once you set up the integral \( \int_{-1}^{2} \sqrt{1 + 4x^2} \, dx \), solving it analytically might be complex.
• Instead, use numerical integration. This method typically involves finding values using rectangles, trapezoids, or more sophisticated algorithms like Simpson's rule.
• Most modern tools like graphing calculators or computer programs will provide these numerical solutions almost instantaneously.
• For our particular problem, the evaluated arc length of the curve \(y = x^2\) comes out to approximately \(5.3884\).

Thus, numerical integration is a valuable technique for approximating complex integrals and making them calculable with current technology.