$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
Is there any normative that regulates the use of silver nanoparticles?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
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?