   Chapter 10, Problem 3RQ

Chapter
Section
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.3. The length of the curve x = f(t), y = g(t), a ≤ t ≤ b, is ∫ a b [ f ′ ( t ) ] 2 + [ g ′ ( t ) ] 2 d t .

To determine

To Find: Whether the given statement is true or false and explain with an example if necessary to prove your answer.

Explanation

Given:

The parametric equation of the variable x is as follows.

x=f(t) (1)

The parametric equation of the variable y is as follows.

y=g(t) (2)

Limits, atb

Calculation:

Consider the example based on the given condition.

Here the equation of the circle is taken.

x2+y2=1(rcosθ)2+(rsin2θ)2=1

r2cos2θ+r2sin2θ=1r2(cos2θ+sin2θ)=1r=1

The is of radius 1 and its length is 2π

For the curve,

x=f(t)=cost (1)

y=g(t)=sint (2)

For 0t4π

Differentiate (1) with respect to t w get,

f'(t)=sint

Differentiate (2) with respect to t w get,

g'(<

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