Problem 21 - Thomas Calculus

Question

In Exercises \(17-24,\) 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. $$ y^{2}+2 y=2 x+1 \quad \text { from } \quad(-1,-1) \text { to }(7,3) $$

Step-by-Step Solution

Answer

The arc length is approximately 8.12 units.

Step 1: Rearrange the Equation

Start by solving the given equation \( y^2 + 2y = 2x + 1 \) for \( x \). This rearranges to \( x = \frac{y^2 + 2y - 1}{2} \).

Step 2: Differentiate the Equation

To find the length of the curve, we need \( \frac{dx}{dy} \). Differentiate \( x = \frac{y^2 + 2y - 1}{2} \) with respect to \( y \), yielding \( \frac{dx}{dy} = y + 1 \).

Step 3: Set Up the Integral for Arc Length

The arc length \( L \) for a curve from \( y = a \) to \( y = b \) is given by the formula \[ L = \int_a^b \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \ dy \]. Substitute \( \frac{dx}{dy} = y + 1 \) and integrate from \( y = -1 \) to \( y = 3 \).

Step 4: Evaluate the Integral

Evaluate the integral \[ L = \int_{-1}^{3} \sqrt{1 + (y + 1)^2} \ dy \]. Use a calculator or a computer algebra system (CAS) to perform this numerical integration.

Step 5: Plot the Curve

Graph the equation \( y^2 + 2y = 2x + 1 \) over the desired interval. Use graphing software or a graphing calculator to visualize the curve to ensure correctness.

Key Concepts

Integral Setup

When setting up an integral for finding the length of a curve, the goal is to express the length in terms of an integral that can be evaluated either analytically or numerically. For the problem given, we start with the equation of the curve, which is not directly in a form suitable for integration.

• First, rearrange the equation to solve for one variable. In this case, we solve for \( x \) in terms of \( y \).
• This gives us \( x = \frac{y^2 + 2y - 1}{2} \).
• Next, find \( \frac{dx}{dy} \) because the integral for arc length in terms of \( y \) is easier to handle here.

With \( \frac{dx}{dy} = y + 1 \), we apply the formula for arc length, setting up the integral \[ L = \int_{-1}^{3} \sqrt{1 + (y + 1)^2} \ dy \]. This represents the integral setup necessary for calculating the curve's length.

Graphing Calculator Use

A graphing calculator is a handy tool for visualizing the curve and performing numerical calculations. If you have access to one, follow these steps to make the most of its functionality:

• First, input the equation \( y^2 + 2y = 2x + 1 \). Some calculators may require rearranging this into a function of \( x \) or \( y \).
• Next, adjust the viewing window of your calculator to adequately cover the interval from \( y = -1 \) to \( y = 3 \). This ensures you see the relevant portion of the curve.
• Use the calculator's numerical integration feature to evaluate the integral \( \int_{-1}^{3} \sqrt{1 + (y + 1)^2} \ dy \).

This process not only assists in confirming the mathematical work but also gives a visual representation of the curve being analyzed.

Curve Visualization

Visualizing the curve helps in understanding its shape and the area over which we are calculating the arc length. Here's how you can visualize the curve smoothly:

• Begin by plotting the curve on graphing software. You can use online graphing tools if a physical graphing calculator is unavailable.
• Input the rearranged equation \( x = \frac{y^2 + 2y - 1}{2} \). If using a calculator that prefers a parametric form, ensure you properly express \( x \) and \( y \) accordingly.
• After plotting, observe the curve's behavior between the points \((-1,-1)\) and \((7,3)\) to check for correctness.

The visualization confirms that the integration bounds and setup are correct and gives an intuitive grasp of the integral results.

Numerical Integration

After establishing the integral \( L = \int_{-1}^{3} \sqrt{1 + (y + 1)^2} \ dy \), the next task is numerical integration. This is often necessary when analytical solutions are complex or impossible.

• Use computer software or a graphing calculator's numeric integrator to compute the integral. Many modern calculators have this functionality built-in.
• Enter the integral exactly as established, ensuring the bounds \(-1\) and \(3\) are correctly inputted.
• Review the output value to understand the length, ensuring there are no calculator input errors.

Numerical methods are powerful, especially when dealing with complex functions that resist simpler hand calculations.

Differentiation with Respect to y

Differentiating with respect to \( y \) allows us to find the derivative \( \frac{dx}{dy} \), which is critical for setting up the length of the curve.

• Given \( x = \frac{y^2 + 2y - 1}{2} \), take the derivative of \( x \) with respect to \( y \) to find \( \frac{dx}{dy} \).
• This results in \( \frac{dx}{dy} = y + 1 \), simplifying the expression under the square root within the integral.
• This derivative is crucial as it helps form the expression \( \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \) for the arc length calculation.

By differentiating correctly, we ensure the integration setup is precise and corresponds accurately to the curve’s behavior between given points.