Arc length with parametric equations, Mathematics

Assignment Help:

Arc Length with Parametric Equations

In the earlier sections we have looked at a couple of Calculus I topics in terms of parametric equations.  We now require to look at a parametric equations.

In this part we will look at the arc length of the parametric curve illustrated by,

x = f (t)

y = g (t)

α ≤ t ≤ β

We will as well be assuming that the curve is traced out exactly one time as t increases from α to β.  We will as well need to suppose that the curve is traced out from left to right as t increases. This is equal to saying,

dx/dt  ≥ 0        for  α ≤ t ≤ β

Thus, let's begin the derivation by recalling the arc length formula since we first derived it in the arc length part of the Applications of Integrals chapter.

L = ∫ ds

In which,

1774_Arc Length with Parametric Equations 2.png

We will make use of the first ds above since we have a nice formula for the derivative in terms of the parametric equations. To make use of this we'll as well need to know that,

 dx = f ′ (t) dt = (dx/dt) dt

After that the arc length formula becomes,

1413_Arc Length with Parametric Equations 3.png

This is a specifically unpleasant formula.  Though, if we factor out the denominator from the square root we reach at,

1816_Arc Length with Parametric Equations 4.png

Here now, utilizing our assumption that the curve is being traced out from left to right we can drop the absolute value bars on the derivative that will permit us to cancel the two derivatives that are outside the square root.


Related Discussions:- Arc length with parametric equations

How to change improper fractions to mixed/ proper fractions, how do you cha...

how do you change an improper fraction to a mixed number or whole or proper

Quadratic equation, find a quadratic equation whose roots are q+1/2 and 2p-...

find a quadratic equation whose roots are q+1/2 and 2p-1 with p+q=1

Rental car agency has 50 cars, Rental car agency has 50 cars. Rental rate i...

Rental car agency has 50 cars. Rental rate in winter is 60%. What is probability that in give winter month the rental rate is fewer than 35 cars rented? Use normal distribution to

Find regular grammar for given regular expression, find regular grammar for...

find regular grammar for the following regular expression: a(a+b)*(ab* +ba*)b

Ronding off numbers, how to round off numbers to the nearest tens and to th...

how to round off numbers to the nearest tens and to the nearest hundred

Example of multiplication, Example 1: Multiply 432 by 8. Solution: ...

Example 1: Multiply 432 by 8. Solution:        432 ×        8 --------------       3,456 In multiplying the multiplier in the units column to the multiplica

Saxon math, what is the are of a square that is 2 inches long and 2 inches...

what is the are of a square that is 2 inches long and 2 inches wide?

Drawn to a circle with center o, From a point P, two tangents PA are drawn ...

From a point P, two tangents PA are drawn to a circle with center O.If OP=diameter of the circle show that triangle APB is equilateral. Ans:    PA=PB (length of tangents

Parallel and perpendicular lines, The last topic that we have to discuss in...

The last topic that we have to discuss in this section is that of parallel & perpendicular lines. Following is a sketch of parallel and perpendicular lines. Suppose that th

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd