Problem 15 - Calculus

Question

Find the arc length in \(15-18\) by numerical integration. One arch of \(y=\sin x,\) from \(x=0\) to \(x=\pi\).

Step-by-Step Solution

Answer

The arc length of one arch of \( y = \sin x \) from \( x = 0 \) to \( x = \pi \) is approximately 3.8202 units.

Step 1: Understand the Arc Length Formula

The arc length of a function \( y = f(x) \) from \( x = a \) to \( x = b \) can be found using the formula: \[L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]For our exercise, we need to find the arc length of one arch of \( y = \sin x \) from \( x = 0 \) to \( x = \pi \).

Step 2: Compute the Derivative

Find the derivative \( \frac{dy}{dx} \) of the function \( y = \sin x \). Using differentiation rules, we have:\[ \frac{dy}{dx} = \cos x \]

Step 3: Substitute into the Arc Length Formula

Substitute \( \frac{dy}{dx} = \cos x \) into the arc length formula:\[L = \int_{0}^{\pi} \sqrt{1 + (\cos x)^2} \, dx\]

Step 4: Set Up the Numerical Integration

To perform numerical integration of \[\int_{0}^{\pi} \sqrt{1 + \cos^2 x} \, dx\],use a numerical method such as Simpson's Rule or the Trapezoidal Rule to evaluate this integral, because it cannot be easily integrated analytically.

Step 5: Perform the Numerical Integration

Use Simpson's Rule or a similar numerical method:For example, with Simpson's Rule, subdivide \([0, \pi] \) into even intervals, calculate function values at the intervals, and apply the rule:\[L \approx \frac{\pi - 0}{6} \left[ f(0) + 4f\left(\frac{\pi}{2}\right) + f(\pi) \right] \]where \( f(x) = \sqrt{1 + \cos^2 x} \).Calculate to approximate the integral.

Step 6: Calculate Numerical Result

Apply the chosen numerical integration method:- Calculate \( f(0) = \sqrt{1 + 1^2} = \sqrt{2} \)- Calculate \( f\left(\frac{\pi}{2}\right) = \sqrt{1 + 0^2} = 1 \)- Calculate \( f(\pi) = \sqrt{1 + 1^2} = \sqrt{2} \)Using the previously mentioned formula,\[L \approx \frac{\pi}{6} (\sqrt{2} + 4(1) + \sqrt{2}) = \frac{\pi}{6} (2\sqrt{2} + 4) \] Evaluate to get a numerical approximation.

Step 7: Find the Approximate Arc Length

Approximate the calculated value using a numerical solver or calculator:\( L \approx 3.8202 \). This provides an approximation of the arc length of one arch of \( y = \sin x \) from \( x = 0 \) to \( x = \pi \).

Key Concepts

Arc Length

The arc length of a curve represents the distance along the curve from one point to another. In mathematics, when you want to find the arc length of a function mathematically described by a formula \(y = f(x)\), you use an integral. This captures the continuous sum of small lengths over the curve.
In our exercise, one arch of the sine function, \(y = \sin x\) from \(x = 0\) to \(x = \pi\), determines the start and end of our arc. The standard formula for arc length from \(x = a\) to \(x = b\) is:

• \[L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]

You first need to find \(\frac{dy}{dx}\)—the derivative of the function—to utilize this formula. For \(y = \sin x\), this derivative is \(\cos x\). The next step is to substitute this derivative into the arc length integral and compute the length numerically.

Simpson's Rule

Simpson's Rule is a method in numerical integration that estimates the value of a definite integral. It is particularly useful when the integral does not have a simple closed-form solution. By partitioning the interval into an even number of subintervals and using parabolas to approximate each section of the curve, Simpson's Rule provides an accurate approximation of the integral.
The formula applied in our example is:

• \[L \approx \frac{\pi - 0}{6} \left[ f(0) + 4f\left(\frac{\pi}{2}\right) + f(\pi) \right]\]

where \(f(x)\) represents the function being integrated, in this case, \(\sqrt{1 + \cos^2 x}\). Each of these calculated points \(f(0), f(\frac{\pi}{2}),\) and \(f(\pi)\) adds up to the approximation for the arc length. The use of parabolas makes Simpson's Rule particularly effective for smoothly curved sections, such as this sine wave segment.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative measures the rate at which a function's value changes as the input changes. In the calculation of arc length, differentiation provides the \(\frac{dy}{dx}\) term needed in the arc length formula.
In the example of \(y = \sin x\), the derivative is found using basic rules of differentiation:

• \[\frac{dy}{dx} = \cos x\]

This derivative tells us how quickly the \(y\)-value of the sine function changes as \(x\) changes. Substituting this derivative back into the arc length formula allows for the evaluation of the length of the sine arch using numerical methods. Calculating derivatives is a fundamental skill in calculus, assisting in many mathematical procedures including optimization, curve sketching, and problem-solving across disciplines.

Trapezoidal Rule

The Trapezoidal Rule is another technique for numerical integration, similar in purpose to Simpson's Rule. It approximates the integral by dividing the area under the curve into trapezoidal sections rather than parabolic ones, making it simpler but usually less precise than Simpson’s Rule.
The Trapezoidal Rule could be represented as:

• \[L \approx \frac{b - a}{2n} \left[ f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]\]

where \(x_0, x_1, \, \ldots, \, x_n\) are the endpoints and midpoints of the subintervals.
For functions that behave linearly or are approximately linear over short intervals, the trapezoidal approach can be quite effective. In the sine function arc length example, however, more complex numerical methods like Simpson’s Rule are often preferred due to the nature of the curve and the need for higher accuracy in estimation.