Q. 15 - Calculus

Question

 By estimating the arc length of $f\left(x\right) = \cos x$ on $[0,\pi ]$in two different approaches, you can compare your results in this exercise.

(a) By applying the distance formula to four line segments.

(b) By approximating the area under the graph of the function $y=\sqrt{1+si{n}^{2}x}$on $[0,\pi ]$ using a right sum with four rectangles.

(c) Why does the arc length of $f\left(x\right) = \cos x$ on $[0,\pi ]$ fall within the range of the approximations you discovered in parts (a) and (b)? Which of the two, if either, do you believe might be a more accurate estimate?

Step-by-Step Solution

Answer

1. The length of the arc of $\cos x$on the range $[0,\pi ].$
2. The region beneath the curve's graph $f\left(x\right)=\sqrt{1+{\sin }^{2}x}$on $[0,\pi ]$ is $3.81994.$
3. It is unclear that of the two approximations provides a more accurate estimate of the arc length.

Part(a) Step 1: Given information

The function $f\left(x\right)= \cos x$$f\left(x\right) = \cos x$

Part(a) Step 2: Calculation

On the range $[0,\pi ]$, sketch the graph of the function $f\left(x\right)=\cos x$. As illustrated in the picture, divide the interval into four equal-width subintervals and draw line segments $AB, BC, CD,$ and $DE$. The subdivision points' coordinates are $(0,1)$,$(\pi /4,0.7071)$,$(\pi /2,0)$,$(3\pi /4,-0.7071)$,$(\pi ,-1)$

Now, the sum of the four line segments is used to approximate the necessary arc length $AE$ along the curve. Thus,

$$\begin{gathered} L=AB+BC+CD+DE \\ =\sqrt{\frac{{\pi }^2}{16}+(0.7071-1{)}^2}+\sqrt{\frac{{\pi }^2}{16}+(0-0.7071{)}^2}+\sqrt{\frac{{\pi }^2}{16}+(-0.7071-0{)}^2} \\ +\sqrt{\frac{{\pi }^2}{16}+(-1+0.7071{)}^2} \\ =2.11362+1.676472 \\ =3.79009 \end{gathered}$$

Consequently, $3.79009$ is roughly the length of the arc of $\cos x$ on the range $[0,\pi ]$.

Part(b) Step 1: Given information

The function $f\left(x\right)=\sqrt{1+{\sin }^{2}x}$ on $[0,\pi ]$.

Part(b) Step 2: Calculation

Draw the graph of the function $f\left(x\right)=\sqrt{1+{\sin }^{2}x}$over the range $[0,\pi ]$, and then divide the range into four equal segments at$\frac{\pi }{4},\frac{\pi }{2},$$\frac{3\pi }{4}$. Therefore, as indicated in the image,$\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4}$ and $\pi$  are the intervals' right end points.

Assess the function's value at these stages now.

$\begin{array}{rcl}f\left(\frac{\pi }{4}\right)& =& 1.22474487\\ f\left(\frac{\pi }{2}\right)& =& 1.41421356\\ f\left(\frac{3\pi }{4}\right)& =& 1.22474487\\ f\left(\pi \right)& =& 1.0\end{array}$

To get the area under the graph of the function $f\left(x\right)$ on an interval$[a, b]$ with $n$ subintervals of equal length$\Delta x$, remember that the right Riemann Sum is provided by the formula.

$A=\Delta x\left[f\right(a+\Delta x)+f(a+2\Delta x)+\dots +f(b\left)\right];$

In the aforementioned relationship $\Delta x=\frac{b-a}{n}$. In this instance $n=4$, so $\Delta x=\frac{\pi -0}{4}=\frac{\pi }{4}$. Use the above-mentioned right sum formula to obtain

$\begin{array}{rcl}A& =& \frac{\pi }{4}[1.22474487+1.41421356+1.22474487+1]\\ & =& \frac{\pi }{4}[4.8637033]\\ & =& 3.81994365\end{array}$

Consequently, the region beneath the curve's graph $f\left(x\right)=\sqrt{1+{\sin }^{2}x}$ on $[0,\pi ]$ is $3.81994.$

Part(c) Step 1: Explanation.

Remember that the definite integral${\int }_a^b\sqrt{1+{\left(f^{\text{'}}\left(x\right)\right)}^2}dx$ gives the precise arc length of a differentiable function $f\left(x\right)$ with continuous derivative on an interval $[a, b]$. This definite integral's value provides the precise arc length. Since the integral was calculated using the Riemann Sum, it only provides an approximation of the value.

Similar to component (a), the method of calculating the arc length using line segments also yields an approximation of the arc length. As a result, only an approximation of the arc length has been estimated in sections (a) and (b). 

The precise length of the arc is unknown since the definite integral cannot be calculated using established integration techniques. As a result, it is uncertain which of the two approximations provides a superior arc length.