TBIL-Calc Arclength (AI2)

Section 6.2 Arclength (AI2)

Learning Outcomes

Estimate the arclength of a curve with Riemann sums and find an integral which computes the arclength.

Subsection 6.2.1 Activities

Activity 6.2.1.

Suppose we wanted to find the arclength of the parabola \(y=-x^2+6x\) over the interval \([0,4]\text{.}\)

Figure 120. Plot of \(y=-x^2+6x\) over \([0,4]\text{.}\)

(a)

Suppose we wished to estimate this length with two line segments where \(\Delta x=2\text{.}\)

Which of the following expressions represents the sum of the lengths of the line segments with endpoints \((0,0)\text{,}\) \((2,8)\) and \((4,8)\text{?}\)

\(\displaystyle \sqrt{4+8}\)
\(\displaystyle \sqrt{2^2+8^2}+\sqrt{(4-2)^2+(8-8)^2}\)
\(\displaystyle \sqrt{4^2+8^2}\)
\(\displaystyle \sqrt{2^2+8^2}+\sqrt{4^2+8^2}\)

(b)

Suppose we wished to estimate this length with four line segments where \(\Delta x=1\text{.}\)

Which of the following expressions represents the sum of the lengths of the line segments with endpoints \((0,0)\text{,}\) \((1,5)\text{,}\) \((2,8)\text{,}\) \((3,9)\) and \((4,8)\text{?}\)

\(\displaystyle \sqrt{4^2+8^2}\)
\(\displaystyle \sqrt{1^2+(5-0)^2}+\sqrt{1^2+(8-5)^2}+\sqrt{1^2+(9-8)^2}+\sqrt{1^2+(8-9)^2}\)
\(\displaystyle \sqrt{1^2+5^2}+\sqrt{2^2+8^2}+\sqrt{3^2+9^2}+\sqrt{4^2+8^2}\)

(c)

Suppose we wished to estimate this length with \(n\) line segments where \(\displaystyle \Delta x=\frac{4}{n}\text{.}\) Let \(f(x)=-x^2+6x\text{.}\)

Which of the following expressions represents the length of the line segment from \((x_0, f(x_0))\) to \((x_0+\Delta x, f(x_0+\Delta x))\text{?}\)

\(\displaystyle \sqrt{x_0^2+f(x_0)^2}\)
\(\displaystyle \sqrt{(x_0+\Delta x)^2+f(x_0+\Delta x)^2}\)
\(\displaystyle \sqrt{(\Delta x)^2+f(\Delta x)^2}\)
\(\displaystyle \sqrt{(\Delta x)^2+(f(x_0+\Delta x)-f(x_0))^2}\)

(d)

Which of the following Riemann sums best estimates the arclength of the parabola \(y=-x^2+6x\) over the interval \([0,4]\text{?}\) Let \(f(x)=-x^2+6x\text{.}\)

\(\displaystyle \displaystyle \sum \sqrt{(\Delta x)^2+f(\Delta x)^2}\)
\(\displaystyle \displaystyle \sum \sqrt{(x_i+\Delta x)^2+f(x_i+\Delta x)^2}\)
\(\displaystyle \displaystyle \sum \sqrt{x_i^2+f(x_i)^2}\)
\(\displaystyle \displaystyle \sum \sqrt{(\Delta x)^2+(f(x_i+\Delta x)-f(x_i))^2}\)

(e)

Note that

\begin{align*} \sqrt{(\Delta x)^2+(f(x_i+\Delta x)-f(x_i))^2} & = \sqrt{(\Delta x)^2\left(1+\left(\frac{f(x_i+\Delta x)-f(x_i)}{\Delta x} \right)^2\right)}\\ &=\sqrt{1+\left(\frac{f(x_i+\Delta x)-f(x_i)}{\Delta x} \right)^2}\Delta x\text{.} \end{align*}

Which of the following best describes \(\displaystyle\lim_{\Delta x\to 0} \frac{f(x_i+\Delta x)-f(x_i)}{\Delta x}\text{?}\)

\(\displaystyle 0\)
\(\displaystyle 1\)
\(\displaystyle f'(x_i)\)
This limit is undefined.

Fact 6.2.2.

Given a differentiable function \(f(x)\text{,}\) the arclength of \(y=f(x)\) defined on \([a,b]\) is computed by the integral

\begin{align*} \lim_{n\to \infty}\sum \sqrt{(\Delta x)^2+(f(x_i+\Delta)-f(x_i))^2} & =\lim_{n\to \infty}\sum \sqrt{1+\left(\frac{f(x_i+\Delta x)-f(x_i)}{\Delta x} \right)^2}\Delta x\\ & = \int_a^b \sqrt{1+(f'(x))^2}dx\text{.} \end{align*}

Activity 6.2.3.

Use Fact 6.2.2 to find an integral which measures the arclength of the parabola \(y=-x^2+6x\) over the interval \([0,4]\text{.}\)

Activity 6.2.4.

Consider the curve \(y=2^x-1\) defined on \([1,5]\text{.}\)

(a)

Estimate the arclength of this curve with two line segments where \(\Delta x=2\text{.}\)

\begin{equation*} \begin{array}{|c|c|c|c|} \hline x_i & (x_i, f(x_i)) & (x_i+\Delta x, f(x_i+\Delta x)) & \text{Length of segment}\\ \hline 1 & & & \\ \hline 3 & & & \\ \hline \end{array} \end{equation*}

(b)

Estimate the arclength of this curve with four line segments where \(\Delta x=1\text{.}\)

\begin{equation*} \begin{array}{|c|c|c|c|} \hline x_i & (x_i, f(x_i)) & (x_i+\Delta x, f(x_i+\Delta x)) & \text{Length of segment}\\ \hline 1 & & & \\ \hline 2 & & & \\ \hline 3 & & & \\ \hline \end{array} \end{equation*}

(c)

Find an integral which computes the arclength of the curve \(y=2^x-1\) defined on \([1,5]\text{.}\)

Activity 6.2.5.

Consider the curve \(y=5e^{-x^2}\) over the interval \([-1,4]\text{.}\)

(a)

Estimate this arclength with 5 line segments where \(\Delta x=1\text{.}\)

(b)

Find an integral which computes this arclength.

Subsection 6.2.2 Videos

Figure 124. Video: Estimate the arclength of a curve with Riemann sums and find an integral which computes the arclength

Calculus for Team-Based Inquiry Learning: 2023 Early Edition

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