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Chapter 9: Q. 1. (page 774)

Fill in the blanks to complete each of the following theorem:

Let C be a curve in the plane with parametrization,y=g(t) for t鈭�a,bsuch that the parametrization is a................... function from the interval a,bto the curve C. If x=f(t)and y=g(t)are differentiable functions of t such that f'(t)and g'(t)are .............on a,b, then the length of the curve C is given by...................

Short Answer

Expert verified

The blanks can be filled in order as:

1. One- to-one

2. Continuous

3.鈭�abf't2+g't2dt

Step by step solution

01

Step 1. Given information

C is the curve

x=f(t)y=g(t)

for t鈭�[a,b]

f'(t),g'(t)are continuous over the interval.

02

Step 2. Write formula for length of the curve.

The Length of the curve in the given interval is:

l=鈭�abdxdt2+dydt2

Since, x=f(t), hence dxdt=f'(t)

y=g(t)dydt=g'(t)

So the length of the curve is:

l=鈭�abf't2+g't2dt

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