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

Identify the flaw in the argument, Identify the flaw in the following argum...

Identify the flaw in the following argument which supposedly determines that n 2 is even when n is an even integer. As well name the reasoning:             Assume that n 2 is

Distinct eigenvalues, It's now time to do solving systems of differential e...

It's now time to do solving systems of differential equations. We've noticed that solutions to the system, x?' = A x? It will be the form of, x? = ?h e l t Here l and

Find the sum of all natural no. between 101 and 304, Find the sum of all na...

Find the sum of all natural no. between 101 & 304 which are divisible by 3 or 5. Find their sum. Ans:    No let 101 and 304, which are divisible by 3. 102, 105..........

Discrete mathmatics, give an example of a relation R that is transitive whi...

give an example of a relation R that is transitive while inverse of R is not

Perimeter of trinagle, what is the perimeter of a triangele with the sides ...

what is the perimeter of a triangele with the sides of 32 in /22 in/20 in/

Determinants, can anyone solve this assigment: D=lsqrt(3x-5) sqrt(2x)l ...

can anyone solve this assigment: D=lsqrt(3x-5) sqrt(2x)l =3 l -1 1 l

Evaluate the area of circle, If the radius of a sphere is doubled, the surf...

If the radius of a sphere is doubled, the surface area is a. multiplied by 4. b. multiplied by 2. c. multiplied by 3. d. multiplied by 8. a. The formula for the surf

What is plotting points, What is Plotting Points ? How would you go abo...

What is Plotting Points ? How would you go about drawing the graph of y = x2 ? One way to do it is by plotting points. (Your graphing calculator uses this method.) This is

What decimal is represented by point a on the number line, What decimal is ...

What decimal is represented by point A on the number line? The hash marks indicate units of 0.01 between 0.75 and 0.80. Point A is 0.77. See the ?gure below.

Center of mass - applications of integrals, Center of Mass - Applications o...

Center of Mass - Applications of integrals In this part we are going to find out the center of mass or centroid of a thin plate along with uniform density ρ. The center of mass

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