Problem 17 - 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. $$ y=x^{2}, \quad-1 \leq x \leq 2 $$

Step-by-Step Solution

Answer

The curve's length is numerically approximated based on the integral setup for the arc length formula.

Step 1: Write the Arc Length Formula

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

Step 2: Substitute into the Arc Length Formula

Substitute \( \frac{dy}{dx} = 2x \) into the arc length formula: \[ L = \int_{-1}^{2} \sqrt{1 + (2x)^2} \, dx = \int_{-1}^{2} \sqrt{1 + 4x^2} \, dx \].

Step 3: Graph the Curve

Plot the curve \( y = x^2 \) within the interval \(-1 \leq x \leq 2\). It should visually resemble a parabola opening upwards, with the vertex falling at the origin and extending from the left at \( x = -1 \) to beyond \( x = 2 \).

Step 4: Evaluate the Integral Numerically

Use a calculator or computer software to evaluate the integral \( \int_{-1}^{2} \sqrt{1 + 4x^2} \, dx \). This will require numerical approximation skills or software capabilities such as a definite integral calculator tool.

Key Concepts

Integral Setup

To find the arc length of a curve, the integral setup is crucial. Starting with the arc length formula for a curve defined by a function \( y = f(x) \) from \( x = a \) to \( x = b \), the formula is: \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]. This formula includes an important derivative component, \( \frac{dy}{dx} \), which captures how the function changes at each point along the curve. For the specific curve \( y = x^2 \), compute the derivative \( \frac{dy}{dx} = 2x \). Next, substitute this derivative back into the formula:\[ L = \int_{-1}^{2} \sqrt{1 + (2x)^2} \, dx = \int_{-1}^{2} \sqrt{1 + 4x^2} \, dx \]. This integral represents the whole arc length we want to find from \( x = -1 \) to \( x = 2 \). Understanding this setup helps us transition smoothly to numerical methods to find the actual length.

Graphing Curves

Seeing the curve visually can provide a better understanding of its shape and important characteristics. For the function \( y = x^2 \), graphing involves plotting this quadratic equation over the interval from \( x = -1 \) to \( x = 2 \).

• The graph will be a parabola that opens upwards.
• This parabola's vertex will be at the origin, which is at the point (0, 0) in the coordinate plane.
• It extends from the left at \( x = -1 \) to the right beyond \( x = 2 \).

A graphing tool or software can be instrumental in visualizing this curve. Managers of visual data can zoom in on specific segments to observe details of how the curve changes. This visualization aids in verifying limits and ensuring that the curve accurately follows the function's rules.

Numerical Integration

Once the integral is set up, numerical integration methods or tools can help find the arc length. This is particularly useful since some integrals, like \( \int_{-1}^{2} \sqrt{1 + 4x^2} \, dx \), cannot be resolved easily by standard analytical methods.

• Computers or calculators with numerical integration capabilities are helpful here.
• They apply computational techniques to approximate the area under the curve represented by the integral.
• The result is a numerical approximation of the arc length.

Methods like Simpson's Rule or the Trapezoidal Rule can be manually employed when software isn't used. Numerical integration is vital for solving complex problems where analytic solutions are intractable, providing a practical approach to estimating measurements.