Therefore, your average test grade is approximately 80.33, which translates to a B− at most schools.
Suppose, however, that we have a function
$v\left(t\right)$ that gives us the speed of an object at any time
t , and we want to find the object’s average speed. The function
$v\left(t\right)$ takes on an infinite number of values, so we can’t use the process just described. Fortunately, we can use a definite integral to find the average value of a function such as this.
Let
$f\left(x\right)$ be continuous over the interval
$\left[a,b\right]$ and let
$\left[a,b\right]$ be divided into
n subintervals of width
$\text{\Delta}x=(b-a)\text{/}n.$ Choose a representative
${x}_{i}^{*}$ in each subinterval and calculate
$f\left({x}_{i}^{*}\right)$ for
$i=1,2\text{,\u2026,}\phantom{\rule{0.2em}{0ex}}n.$ In other words, consider each
$f\left({x}_{i}^{*}\right)$ as a sampling of the function over each subinterval. The average value of the function may then be approximated as
Following through with the algebra, the numerator is a sum that is represented as
$\sum _{i=1}^{n}f\left({x}_{i}^{*}\right)},$ and we are dividing by a fraction. To divide by a fraction, invert the denominator and multiply. Thus, an approximate value for the average value of the function is given by
Let
$f\left(x\right)$ be continuous over the interval
$\left[a,b\right].$ Then, the
average value of the function$f\left(x\right)$ (or
f_{ave} ) on
$\left[a,b\right]$ is given by
Find the average value of
$f\left(x\right)=x+1$ over the interval
$\left[0,5\right].$
First, graph the function on the stated interval, as shown in
[link] .
The region is a trapezoid lying on its side, so we can use the area formula for a trapezoid
$A=\frac{1}{2}h\left(a+b\right),$ where
h represents height, and
a and
b represent the two parallel sides. Then,
The definite integral can be used to calculate net signed area, which is the area above the
x -axis less the area below the
x -axis. Net signed area can be positive, negative, or zero.
The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration.
Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities.
The properties of definite integrals can be used to evaluate integrals.
The area under the curve of many functions can be calculated using geometric formulas.
The average value of a function can be calculated using definite integrals.
Key equations
Definite Integral ${\int}_{a}^{b}f\left(x\right)dx}=\underset{n\to \infty}{{\displaystyle \text{lim}}}{\displaystyle \sum _{i=1}^{n}f\left({x}_{i}^{*}\right)\text{\Delta}x$
Properties of the Definite Integral ${\int}_{a}^{a}f\left(x\right)dx=0$ ${\int}_{b}^{a}f\left(x\right)dx}=\text{\u2212}{\displaystyle {\int}_{a}^{b}f\left(x\right)dx$ ${\int}_{a}^{b}\left[f\left(x\right)+g\left(x\right)\right]dx}={\displaystyle {\int}_{a}^{b}f\left(x\right)dx}+{\displaystyle {\int}_{a}^{b}g\left(x\right)dx$ $\int}_{a}^{b}\left[f\left(x\right)-g\left(x\right)\right]dx={\displaystyle {\int}_{a}^{b}f\left(x\right)dx-{\displaystyle {\int}_{a}^{b}g\left(x\right)dx}$ ${\int}_{a}^{b}cf\left(x\right)dx}=c{\displaystyle {\int}_{a}^{b}f\left(x\right)$ for constant
c ${\int}_{a}^{b}f\left(x\right)dx}={\displaystyle {\int}_{a}^{c}f\left(x\right)dx}+{\displaystyle {\int}_{c}^{b}f\left(x\right)dx$
Questions & Answers
can someone help me with some logarithmic and exponential equations.
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
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.