This speed translates to approximately 95 mph—a major-league fastball.
Surface area generated by a parametric curve
Recall the problem of finding the surface area of a volume of revolution. In
Curve Length and Surface Area , we derived a formula for finding the surface area of a volume generated by a function
$y=f\left(x\right)$ from
$x=a$ to
$x=b,$ revolved around the
x -axis:
We now consider a volume of revolution generated by revolving a parametrically defined curve
$x=x\left(t\right),y=y\left(t\right),a\le t\le b$ around the
x -axis as shown in the following figure.
The analogous formula for a parametrically defined curve is
The derivative of the parametrically defined curve
$x=x\left(t\right)$ and
$y=y\left(t\right)$ can be calculated using the formula
$\frac{dy}{dx}=\frac{{y}^{\prime}(t)}{{x}^{\prime}(t)}.$ Using the derivative, we can find the equation of a tangent line to a parametric curve.
The area between a parametric curve and the
x -axis can be determined by using the formula
$A={\displaystyle {\int}_{{t}_{1}}^{{t}_{2}}y\left(t\right){x}^{\prime}\left(t\right)\phantom{\rule{0.2em}{0ex}}dt}.$
The arc length of a parametric curve can be calculated by using the formula
$s={\displaystyle {\int}_{{t}_{1}}^{{t}_{2}}\sqrt{{\left(\frac{dx}{dt}\right)}^{2}+{\left(\frac{dy}{dt}\right)}^{2}}dt}.$
The surface area of a volume of revolution revolved around the
x -axis is given by
$S=2\pi {\displaystyle {\int}_{a}^{b}y\left(t\right)\sqrt{{\left({x}^{\prime}\left(t\right)\right)}^{2}+{\left({y}^{\prime}\left(t\right)\right)}^{2}}dt}.$ If the curve is revolved around the
y -axis, then the formula is
$S=2\pi {\displaystyle {\int}_{a}^{b}x\left(t\right)\sqrt{{\left({x}^{\prime}\left(t\right)\right)}^{2}+{\left({y}^{\prime}\left(t\right)\right)}^{2}}dt}.$
Key equations
Derivative of parametric equations $\frac{dy}{dx}=\frac{dy\text{/}dt}{dx\text{/}dt}=\frac{{y}^{\prime}\left(t\right)}{{x}^{\prime}\left(t\right)}$
Second-order derivative of parametric equations $\frac{{d}^{2}y}{d{x}^{2}}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{\left(d\text{/}dt\right)\left(dy\text{/}dx\right)}{dx\text{/}dt}$
Area under a parametric curve $A={\displaystyle {\int}_{a}^{b}y\left(t\right){x}^{\prime}\left(t\right)\phantom{\rule{0.2em}{0ex}}dt}$
Arc length of a parametric curve $s={\displaystyle {\int}_{{t}_{1}}^{{t}_{2}}\sqrt{{\left(\frac{dx}{dt}\right)}^{2}+{\left(\frac{dy}{dt}\right)}^{2}}dt}$
Surface area generated by a parametric curve $S=2\pi {\displaystyle {\int}_{a}^{b}y\left(t\right)\sqrt{{\left({x}^{\prime}\left(t\right)\right)}^{2}+{\left({y}^{\prime}\left(t\right)\right)}^{2}}dt}$
For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.
Questions & Answers
Do somebody tell me a best nano engineering book for beginners?
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.
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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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?