Q. 25 Find the arc length of the curve... [FREE SOLUTION]

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Chapter 11: Q. 25 (page 889)

Find the arc length of the curves defined by the vector-valued functions on the specified intervals in Exercises 22鈥�27.
$r (t) = (4 \sin t, t^{3 / 2}, - 4 \cos t), [0, 4]$

Short Answer

Expert verified

The arc length of curve $r (t) = <4 \sin t, t^{\frac{3}{2}}, - 4 \cos t> on [0, 4] is \frac{488}{27}$ .

Step by step solution

Step 1. Given information.

The given curve is $(t) = <4 \sin t, t^{\frac{3}{2}}, - 4 \cos t> on [0, 4] .$

Step 2. Arc-Length.

The arc length is given by,

$I (a, b) = {鈭�}_{a}^{b} ||r^{'} (t)|| d t r^{'} (t) = <4 \cos t, \frac{3}{2} t^{\frac{1}{2}}, 4 \sin t> ||r^{'} (t)|| = \sqrt{(4 \cos t)^{2} + {(\frac{3}{2} \sqrt{t})}^{2} + (4 \sin t)^{2}} = \sqrt{16 (\cos^{2} t + \sin^{2} t) + \frac{9}{4} t} = \frac{1}{2} \sqrt{64 + 9 t} N o w, (l) = {鈭�}_{a}^{b} ||r^{'} (t)|| d t = {鈭�}_{0}^{4} \frac{1}{2} \sqrt{64 + 9 t} d t {= \frac{1}{2} \frac{(64 + 9 t)^{\frac{3}{2}}}{\frac{3}{2}} 路 \frac{1}{9}]}_{0}^{4} = \frac{1}{27} [(64 + 9 (4))^{\frac{3}{2}} - (64 + 9 (0))^{\frac{3}{2}}] = \frac{1}{27} [(100)^{\frac{3}{2}} - (64)^{\frac{3}{2}}] = \frac{1}{27} [1000 - 512] = \frac{488}{27}$

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91影视

Find the arc length of the curves defined by the vector-valued functions on the specified intervals in Exercises 22鈥�27.
$r (t) = (4 \sin t, t^{3 / 2}, - 4 \cos t), [0, 4]$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Arc-Length.

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91影视

Find the arc length of the curves defined by the vector-valued functions on the specified intervals in Exercises 22鈥�27.r(t)=4sint,t3/2,-4cost,[0,4]

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Arc-Length.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Calculus

Decision Maths

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Find the arc length of the curves defined by the vector-valued functions on the specified intervals in Exercises 22鈥�27.
$r (t) = (4 \sin t, t^{3 / 2}, - 4 \cos t), [0, 4]$