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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
\([0,4]\text{.}\)
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
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X
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W
Extra details if available
Space
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M
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Comma
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N
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Z
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C
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T
Monochrome colours
L
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K
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Y
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O
Start and stop sonification
P
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Plot of \(y=-x^2+6x\) over \([0,4]\text{.}\)
(a) Suppose we wished to estimate this length with two line segments where
\(\Delta x=2\text{.}\)
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Plot of
\(y=-x^2+6x\) over
\([0,4]\) with a piecewise linear approximation consisting of two pieces.
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
\(\Delta x=1\text{.}\)
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Plot of
\(y=-x^2+6x\) over
\([0,4]\) with a piecewise linear approximation consisting of four pieces.
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
\(f(x)=-x^2+6x\text{.}\)
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
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
\(\Delta x=2\text{.}\)
\(x_i\) \((x_i,f(x_i))\) \((x_i+\Delta x,f(x_i+\Delta x))\) Length of segment
\(1\)
\(3\)
(b) Estimate the arclength of this curve with four line segments where
\(\Delta x=1\text{.}\)
\(x_i\) \((x_i,f(x_i))\) \((x_i+\Delta x,f(x_i+\Delta x))\) Length of segment
\(1\)
\(2\)
\(3\)
\(4\)
(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 88. Video: Estimate the arclength of a curve with Riemann sums and find an integral which computes the arclength
Subsection 6.2.3 Exercises
