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. $$y=4 x-x^{2} ;[0,4]$$

Short answer

Answer: The approximate arc length of the curve is 7.42.

Step-by-step solution

Write the Integral for the Arc Length

Recall the arc length formula for a curve defined by the function $$y = f(x)$$ on the interval $$[a, b]$$: $$L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} \, dx$$ First, we need to find the derivative of the given function, $$f(x) = 4x - x^2$$. $$f'(x) = 4 - 2x$$ Now, we can plug this into the arc length formula: $$L = \int_{0}^{4} \sqrt{1 + (4 - 2x)^2} \, dx$$

Simplify the Integral

Square the derivative inside the square root: $$L = \int_{0}^{4} \sqrt{1 + (16 - 16x + 4x^2)} \, dx$$ Now, simplify the terms inside the square root: $$L = \int_{0}^{4} \sqrt{1 + 4x^2 - 16x + 16} \, dx$$

Use Technology to Evaluate or Approximate the Integral

Now that we have simplified the integral, we can use technology (such as a graphing calculator or a computer algebra system) to evaluate or approximate the integral. Use a calculator or computer software to input our integral: $$L = \int_{0}^{4} \sqrt{1 + 4x^2 - 16x + 16} \, dx$$ This yields an approximate value of 7.42 (rounded to two decimal places) for the arc length. So, the arc length of the curve $$y = 4x - x^2$$ on the interval $$[0, 4]$$ is approximately 7.42.

Key concepts

Arc Length Formula

To find the arc length of a function on a certain interval, we use the arc length formula. The formula is essential because it calculates the length of the curve precisely. This is particularly useful in calculus when the path length of a curve needs to be determined between two points. For a function \( y = f(x) \) defined over an interval \([a, b]\), the arc length \( L \) is calculated by:\[ L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} \, dx \]This formula is derived from the Pythagorean theorem and considers the infinitesimal triangles along the curve. The line over which the curve stretches is segmented into tiny portions, and the formula integrates these segments from the start to the end of the interval. The expression \( (f'(x))^2 \) represents the square of the derivative of the function, indicating how steep the curve is at each point.

Integral Simplification

Once the arc length integral is set up, the next task is to simplify it. Integral simplification helps to make the integration process easier, particularly when dealing with complex expressions. Generally, this involves algebraic manipulation to simplify the expression under the square root sign in the integral.For example, in the given arc length integral:\[ L = \int_{0}^{4} \sqrt{1 + (4 - 2x)^2} \, dx \]The derivative inside the square root is squared, giving:\[ (4 - 2x)^2 = 16 - 16x + 4x^2 \]This expression can be combined with 1, as:\[ 1 + 16 - 16x + 4x^2 = 4x^2 - 16x + 17 \]Simplification can be the key step that makes the integration either feasible by hand or more suited for numerical methods or technology.

Derivative Calculation

Calculating derivatives is a crucial step in setting up the arc length integral. It provides insight into how the function behaves along its domain. The derivative \( f'(x) \) indicates the slope or rate of change of a function at any given point \( x \).For the function \( y = 4x - x^2 \), the derivative is calculated as follows:- The derivative of \( 4x \) is 4.- The derivative of \( -x^2 \) is \(-2x\).Thus, the derivative \( f'(x) \) is:\[ f'(x) = 4 - 2x \]This derivative is then used in the arc length formula to compute \( (f'(x))^2 \). Proper calculation of this derivative is vital since it affects the eventual integration process and the accuracy of the arc length.

Numerical Integration

When an integral is too complex to solve analytically, as often happens with arc lengths, numerical integration becomes the go-to method. This approach uses algorithms for approximating the value of an integral, essentially by summing the areas of shapes (such as rectangles) that approximate the region under the curve.Several numerical methods available include:

• Trapezoidal Rule
• Simpson's Rule
• Monte Carlo Integration

For the arc length:\[ L = \int_{0}^{4} \sqrt{1 + 4x^2 - 16x + 16} \, dx \],using technology or software simplifies the process greatly. On plugging in this integral into a calculator, we approximate the arc length as about 7.42. Such tools allow students and professionals to handle otherwise complex integrals with straightforward computational power, making numerical integration a powerful asset in calculus.