   Chapter 13.3, Problem 4E

Chapter
Section
Textbook Problem

# Find the length of the curve.4. r(t) =cos t i + sin t j +ln cos t k, 0 ≤ t ≤ π/4

To determine

To find: The length of the curve L for the vector equation r(t)=costi+sintj+lncostk,0tπ4 .

Explanation

Given data:

r(t)=costi+sintj+lncostk,0tπ4

Formula used:

Write the expression to find length of the curve L for the vector r(t) .

L=ab|r(t)|dt (1)

Here,

r(t) is the tangent vector, which is the derivative of vector r(t) , and

[a,b] is parameter interval.

Find the tangent vector r(t) by differentiating each component of the vector r(t) as follows.

ddt[r(t)]=ddt[costi+sintj+lncostk]

r(t)=ddt(costi)+ddt(sintj)+ddt(lncostk) (2)

Write the following formula to compute the expression for r(t) .

ddt(cost)=sintddt(sint)=cost

Apply chain rule and write the formula to find r(t) .

ddt(lncost)=[ddt(lnu)]dudt{u=costddt(lnu)=1u}=[1u]dudt=1costd(cost)dt=sintcost

ddt(lncost)=tant

Substitute sint for ddt(cost) , cost for ddt(sint) , and tant for ddt(lncost) in equation (2),

r(t)=sinti+costjtantk (3)

Take magnitude on both sides of equation (3)

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