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Calculus for Team-Based Inquiry Learning: PREVIEW Edition — Instructor Version

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Section 6.2 Arclength (AI2)

Subsection 6.2.1 Activities

Activity 6.2.1.

Suppose we wanted to find the arclength of the parabola y=−x2+6x over the interval [0,4].
described in detail following the image
Plot of y=−x2+6x over [0,4].
Figure 126. Plot of y=−x2+6x over [0,4].
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(a)
Suppose we wished to estimate this length with two line segments where Δx=2.
described in detail following the image
Plot of y=−x2+6x over [0,4].
Figure 127. Plot of y=−x2+6x over [0,4] with two line segments where Δx=2.
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Which of the following expressions represents the sum of the lengths of the line segments with endpoints (0,0), (2,8) and (4,8)?
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  1. 4+8
  2. 22+82+(4−2)2+(8−8)2
  3. 42+82
  4. 22+82+42+82
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(b)
Suppose we wished to estimate this length with four line segments where Δx=1.
described in detail following the image
Plot of y=−x2+6x over [0,4].
Figure 128. Plot of y=−x2+6x over [0,4] with four line segments where Δx=1.
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Which of the following expressions represents the sum of the lengths of the line segments with endpoints (0,0), (1,5), (2,8), (3,9) and (4,8)?
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  1. 42+82
  2. 12+(5−0)2+12+(8−5)2+12+(9−8)2+12+(8−9)2
  3. 12+52+22+82+32+92+42+82
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(c)
Suppose we wished to estimate this length with n line segments where Δx=4n. Let f(x)=−x2+6x.
described in detail following the image
Plot of y=−x2+6x over [0,4].
Figure 129. Plot of y=−x2+6x over [0,4] with n line segments where Δx=4n.
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Which of the following expressions represents the length of the line segment from (x0,f(x0)) to (x0+Δx,f(x0+Δx))?
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  1. x02+f(x0)2
  2. (x0+Δx)2+f(x0+Δx)2
  3. (Δx)2+f(Δx)2
  4. (Δx)2+(f(x0+Δx)−f(x0))2
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(d)
Which of the following Riemann sums best estimates the arclength of the parabola y=−x2+6x over the interval [0,4]? Let f(x)=−x2+6x.
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  1. ∑(Δx)2+f(Δx)2
  2. ∑(xi+Δx)2+f(xi+Δx)2
  3. ∑xi2+f(xi)2
  4. ∑(Δx)2+(f(xi+Δx)−f(xi))2
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(e)
Note that
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(Δx)2+(f(xi+Δx)−f(xi))2=(Δx)2(1+(f(xi+Δx)−f(xi)Δx)2)=1+(f(xi+Δx)−f(xi)Δx)2Δx.
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Which of the following best describes limΔx→0f(xi+Δx)−f(xi)Δx?
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  1. 0
  2. 1
  3. f′(xi)
  4. This limit is undefined.
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Activity 6.2.3.

Use Fact 6.2.2 to find an integral which measures the arclength of the parabola y=−x2+6x over the interval [0,4].
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Activity 6.2.4.

Consider the curve y=2x−1 defined on [1,5].
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(a)
Estimate the arclength of this curve with two line segments where Δx=2.
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xi (xi,f(xi)) (xi+Δx,f(xi+Δx)) Length of segment
1
3
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(b)
Estimate the arclength of this curve with four line segments where Δx=1.
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xi (xi,f(xi)) (xi+Δx,f(xi+Δx)) Length of segment
1
2
3
4
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(c)
Find an integral which computes the arclength of the curve y=2x−1 defined on [1,5].
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Activity 6.2.5.

Consider the curve y=5e−x2 over the interval [−1,4].
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(a)
Estimate this arclength with 5 line segments where Δx=1.
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(b)
Find an integral which computes this arclength.
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Subsection 6.2.2 Videos

Figure 130. Video: Estimate the arclength of a curve with Riemann sums and find an integral which computes the arclength
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Subsection 6.2.3 Exercises

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