$A=3\pi $ (Note that the integral formula actually yields a negative answer. This is due to the fact that
$x\left(t\right)$ is a decreasing function over the interval
$\left[0,2\pi \right];$ that is, the curve is traced from right to left.)
In addition to finding the area under a parametric curve, we sometimes need to find the arc length of a parametric curve. In the case of a line segment, arc length is the same as the distance between the endpoints. If a particle travels from point
A to point
B along a curve, then the distance that particle travels is the arc length. To develop a formula for arc length, we start with an approximation by line segments as shown in the following graph.
Given a plane curve defined by the functions
$x=x\left(t\right),y=y\left(t\right),a\le t\le b,$ we start by partitioning the interval
$[a,b]$ into
n equal subintervals:
${t}_{0}=a<{t}_{1}<{t}_{2}<\text{\cdots}<{t}_{n}=b.$ The width of each subinterval is given by
$\text{\Delta}t=(b-a)\text{/}n.$ We can calculate the length of each line segment:
If we assume that
$x\left(t\right)$ and
$y\left(t\right)$ are differentiable functions of
t, then the Mean Value Theorem (
Introduction to the Applications of Derivatives ) applies, so in each subinterval
$[{t}_{k-1},{t}_{k}]$ there exist
${\hat{t}}_{k}$ and
${\tilde{t}}_{k}$ such that
This is a Riemann sum that approximates the arc length over a partition of the interval
$[a,b].$ If we further assume that the derivatives are continuous and let the number of points in the partition increase without bound, the approximation approaches the exact arc length. This gives
When taking the limit, the values of
${\hat{t}}_{k}$ and
${\tilde{t}}_{k}$ are both contained within the same ever-shrinking interval of width
$\text{\Delta}t,$ so they must converge to the same value.
Questions & Answers
where we get a research paper on Nano chemistry....?
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest.
Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.?
How this robot is carried to required site of body cell.?
what will be the carrier material and how can be detected that correct delivery of drug is done
Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?