Area with parametric equations - polar coordinates, Mathematics

Assignment Help:

Area with Parametric Equations

In this section we will find out a formula for ascertaining the area under a parametric curve specified by the parametric equations,

x = f (t)

y = g (t)

We will as well need to further add in the assumption that the curve is traced out precisely once as t increases from α to β.

We will do this in much similar way that we found the first derivative in the preceding section.

We will first remind how to find out the area under y = F(x) on a < x < b.

A = ∫ba F (x) dx

We will here think of the parametric equation x = f (t) as a substitution in the integral. We will as well assume that a = f(α) and b=f (β)) for the purposes of this formula.  There is in fact no reason to assume that this will always be the case and so we will provide a corresponding formula later if it's the opposed case (b = f (α) and a = f (β)).

Thus, if this is going to be a substitution we'll require,

dx = f' (t) dt

Plugging this into the area formula on top of and making sure to change the limits to their corresponding t values provides us,

A = ∫βα F (f (t)) f' (t) dt

As we don't know what F(x) is we'll use the fact that

y = F (x)

= F (f (t)) = g (t)

and we reach at the formula that we want.

 Area under Parametric Curve, Formula I

A = ∫βα g(t) f' (t) dt

Now, if we should happen to have b = f (α) and a = f (β) then the formula would be,

Area Under Parametric Curve, Formula II

A = ∫βα g(t) f' (t) dt


Related Discussions:- Area with parametric equations - polar coordinates

Question, What is a marketing plan

What is a marketing plan

Need answer urgently, using a pair of compasses a ruler and a pencil. const...

using a pair of compasses a ruler and a pencil. construct a triangle CDE in which DE=10cm, DC+8cm and CDE= 45 degrees. construct CF perpendicular to DE such that F lies on DE using

Ratio, which ratio is largar. 1. 15:16 or 24:25

which ratio is largar. 1. 15:16 or 24:25

The volume and surface area of this solid , The region bounded by y=e -x a...

The region bounded by y=e -x and the x-axis among x = 0 and x = 1 is revolved around the x-axis. Determine the volume and surface area of this solid of revolution.

Developing an understanidng of multiplication, DEVELOPING AN UNDERSTANIDNG ...

DEVELOPING AN UNDERSTANIDNG OF MULTIPLICATION :  The most important aspect of knowing multiplication is to understand what it means and where it is applied. It needs to be first i

Tent originally sold for $2 what is the percent of discount, A tent origina...

A tent originally sold for $260 and has been marked down to $208. What is the percent of discount? Find out the number of dollars off. $260 - $208 = $52. Further, determine wha

Number sentences, when i couulate the formula f 64 divided by 65 how do i d...

when i couulate the formula f 64 divided by 65 how do i do this

Thinking mathematically-why learn mathematics, THINKING MATHEMATICALLY :  ...

THINKING MATHEMATICALLY :  Have you ever thought of what mental processes you are going through when you are solving a mathematical problem? Why don't you try the following proble

Determine the tangent line to f ( x ) = 15 - 2x2 at x = 1, Determine the t...

Determine the tangent line to f ( x ) = 15 - 2x 2   at x = 1. Solution : We know from algebra that to determine the equation of a line we require either two points onto the li

Tutor, how can i apply as tutor

how can i apply as tutor

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