Exercises

In Exercises 1–6, compute the length of the curve over the given interval.

Question 13.137

Question 13.138

Question 13.139

Question 13.140

Question 13.141

Question 13.142

. Use the formula:

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In Exercises 7 and 8, compute the arc length function for the given value of a.

Question 13.143

Question 13.144

In Exercises 9–12, find the speed at the given value of t.

Question 13.145

Question 13.146

Question 13.147

Question 13.148

Question 13.149

What is the velocity vector of a particle traveling to the right along the hyperbola y = x−1 with constant speed 5 cm/s when the particle’s location is ?

Question 13.150

A bee with velocity vector r′(t) starts out at the origin at t = 0 and flies around for T seconds. Where is the bee located at time T if ? What does the quantity represent?

Question 13.151

Let

  • Show that r(t) parametrizes a helix of radius R and height h making N complete turns.

  • Guess which of the two springs in Figure 5 uses more wire.

    Which spring uses more wire?

  • Compute the lengths of the two springs and compare.

Question 13.152

Use Exercise 15 to find a general formula for the length of a helix of radius R and height h that makes N complete turns.

Question 13.153

The cycloid generated by the unit circle has parametrization

r(t) = 〈t − sin t, 1 − cos t〉

  • Find the value of t in [0, 2π] where the speed is at a maximum.

  • Show that one arch of the cycloid has length 8. Recall the identity sin2(t/2) = (1 − cos t)/2.

Question 13.154

Which of the following is an arc length parametrization of a circle of radius 4 centered at the origin?

  • r1(t) = 〈4 sin t, 4 cos t〉

  • r2(t) = 〈4 sin 4t, 4 cos 4t〉

Question 13.155

Let r(t) = 〈3t + 1, 4t − 5, 2t〉.

  • Evaluate the arc length integral .

  • Find the inverse g(s) of s(t).

  • Verify that r1(s) = r(g(s)) is an arc length parametrization.

Question 13.156

Find an arc length parametrization of the line y = 4x + 9.

Question 13.157

Let r(t) = w + tv be the parametrization of a line.

  • Show that the arc length function is given by s(t) = t∥v∥. This shows that r(t) is an arc length parametrizaton if and only if v is a unit vector.

  • Find an arc length parametrization of the line with w = 〈1, 2, 3〉 and v = 〈3, 4, 5〉.

Question 13.158

Find an arc length parametrization of the circle in the plane z = 9 with radius 4 and center (1, 4, 9).

Question 13.159

Find a path that traces the circle in the plane y = 10 with radius 4 and center (2, 10, −3) with constant speed 8.

Question 13.160

Find an arc length parametrization of r(t) = 〈et sin t, et cos t, et 〉.

Question 13.161

Find an arc length parametrization of r(t) = 〈t2, t3〉.

Question 13.162

Find an arc length parametrization of the cycloid with parametrization r(t) = 〈t − sin t, 1 − cos t〉.

Question 13.163

Find an arc length parametrization of the line y = mx for an arbitrary slope m.

Question 13.164

Express the arc length L of y = x3 for 0 ≤ x ≤ 8 as an integral in two ways, using the parametrizations r1(t) = 〈t, t3〉 and r2(t) = 〈t3, t9〉. Do not evaluate the integrals, but use substitution to show that they yield the same result.

Question 13.165

The curve known as the Bernoulli spiral (Figure 6) has parametrization r(t) = 〈et cos 4t, et sin 4t〉.

  • Evaluate . It is convenient to take lower limit −∞ because r(−∞) = 〈0, 0〉.

  • Use (a) to find an arc length parametrization of r(t).

Bernoulli spiral.

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