5128_Ch07_pp378-433

30. Because the limit of the sum x k as the norm of the partition goes to
zero will always be the length (b a) of the interval (a, b).
y
y 25 cos (x/50)
Section 7.4 Lengths of Curves 417
34. Multiple Choice Which of the following gives the best
approximation of the length of the arc of y cos(2x) from x 0
to x p/4? D
(A) 0.785 (B) 0.955 (C) 1.0 (D) 1.318 (E) 1.977
–25
0
25
NOT TO SCALE
300 ft
x (ft)
35. Multiple Choice Which of the following expressions gives
the length of the graph of x y 3 from y 2 to y 2? C
2
(A)2
2
(1 y 6 ) dy (B)
2
(C) 1 9y 4 dy
2
2
(E) 1 x 4 dx
2
2
1 y 6 dy
2
(D) 1 x 2 dx
2
36. Multiple Choice Find the length of the curve described by
y 2 3 x3/2 from x 0 to x 8.
B
(A) 2 6
(B) 5 2
(C) 512 2
3
3
15
(D) 512 2
8 (E) 96
15
In Exercises 25 and 26, find the length of the curve.
25. f x x 13 x 23 , 0 x 2 3.6142
x 1
26. f x 4 x
2 , 1 1 2 x 1 2.1089
In Exercises 27–29, find the length of the nonsmooth curve.
27. y x 3 5x from x 2 to x 1 13.132
28. x y 1 1.623
29. y 4 x from x 0 to x 16 16.647
30. Writing to Learn Explain geometrically why it does not work
to use short horizontal line segments to approximate the lengths
of small arcs when we search for a Riemann sum that leads to the
formula for arc length.
31. Writing to Learn A curve is totally contained inside the
square with vertices 0, 0, 1, 0, 1, 1, and 0, 1. Is there any
limit to the possible length of the curve? Explain.
Standardized Test Questions
You should solve the following problems without using
a graphing calculator.
32. True or False If a function y f (x) is continuous on an
interval [a, b], then the length of its curve is given by
b
1 d y a
d
x 2
dx. Justify your answer.
False. The function must be differentiable.
33. True or False If a function y f (x) is differentiable on an
interval [a, b], then the length of its curve is given by
b
y x 2
dx. Justify your answer.
True, by definition.
a
1 d d
31. No. Consider the curve y 1 3 sin 1 x
0.5 for 0 x 1.
37. Multiple Choice Which of the following expressions should
be used to find the length of the curve y x 2/3 from x 1
to x 1? A
1
(A) 2 1 9 0
4 y
1
(C)0
1
(E)
0
1 y 3 dy
1 y 9/4
dy
Exploration
1
dy (B)1 1 9 4 y
dy
1
(D) 1 y 6 dy
0
38. Modeling Running Tracks Two lanes of a running track
are modeled by the semiellipses as shown. The equation for
lane 1 is y 100 0.2x 2 , and the equation for lane 2
is y 15 0 0.2x 2 . The starting point for lane 1 is at the
negative x-intercept 50 0,0. The finish points for both lanes
are the positive x-intercepts. Where should the starting point be
placed on lane 2 so that the two lane lengths will be equal
(running clockwise)? (–19.909, 8.410)
Start lane 2
Start lane 1
1
y
10
x