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Chapter 11: Q. 26 (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)=et,2t,e-t,[0,1]

Short Answer

Expert verified

The arc-length of the curver(t)=et,2t,e-ton[0,1]ise2-1e.

Step by step solution

01

Step 1. Given information.

The given curve isr(t)=et,2t,e-ton[0,1].

02

Step 2. Arc-length.

The arc-length is given by,

I(a,b)=鈭�abr'(t)dtr'(t)=et2+(2)2+-et2=e2t+2+1e2t=e4t+2e2t+1et=e2t+1et=et+e-t(l)=鈭�abr'(t)dt=鈭�01et+e-tdt=et-e-t01=e1-e-1-e0-e-0=e-1e-(1-1)=e2-1e

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Most popular questions from this chapter

Let r(t)=x(t),y(t),t鈭�[a,鈭�),be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a horizontal asymptote as t鈫�鈭�?Provide an example illustrating your answer.

For each of the vector-valued functions in Exercises, find the unit tangent vector and the principal unit normal vector at the specified value of t.

rt=sin2t,cos2t,t,t=蟺4

For each of the vector-valued functions in Exercises 22鈥�28, find the unit tangent vector.

r(t)=(cos3t,sin3t)

For each of the vector-valued functions, find the unit tangent vector.

r(t)=(t,t2,t3)

Find parametric equations for each of the vector-valued functions in Exercises 26鈥�34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

r(t)=(cos2t,4int,t)fort鈭�[0,2蟺]
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