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89 + 90 + 56 + 78 + 100 + 69 6 = 482 6 80.33 .

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 ( t ) 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 ( t ) 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 ( x ) be continuous over the interval [ a , b ] and let [ a , b ] be divided into n subintervals of width Δ x = ( b a ) / n . Choose a representative x i * in each subinterval and calculate f ( x i * ) for i = 1 , 2 ,…, n . In other words, consider each f ( x i * ) as a sampling of the function over each subinterval. The average value of the function may then be approximated as

f ( x 1 * ) + f ( x 2 * ) + + f ( x n * ) n ,

which is basically the same expression used to calculate the average of discrete values.

But we know Δ x = b a n , so n = b a Δ x , and we get

f ( x 1 * ) + f ( x 2 * ) + + f ( x n * ) n = f ( x 1 * ) + f ( x 2 * ) + + f ( x n * ) ( b a ) Δ x .

Following through with the algebra, the numerator is a sum that is represented as i = 1 n f ( x i * ) , 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

i = 1 n f ( x i * ) ( b a ) Δ x = ( Δ x b a ) i = 1 n f ( x i * ) = ( 1 b a ) i = 1 n f ( x i * ) Δ x .

This is a Riemann sum. Then, to get the exact average value, take the limit as n goes to infinity. Thus, the average value of a function is given by

1 b a lim n i = 1 n f ( x i ) Δ x = 1 b a a b f ( x ) d x .


Let f ( x ) be continuous over the interval [ a , b ] . Then, the average value of the function f ( x ) (or f ave ) on [ a , b ] is given by

f ave = 1 b a a b f ( x ) d x .

Finding the average value of a linear function

Find the average value of f ( x ) = x + 1 over the interval [ 0 , 5 ] .

First, graph the function on the stated interval, as shown in [link] .

A graph in quadrant one showing the shaded area under the function f(x) = x + 1 over [0,5].
The graph shows the area under the function f ( x ) = x + 1 over [ 0 , 5 ] .

The region is a trapezoid lying on its side, so we can use the area formula for a trapezoid A = 1 2 h ( a + b ) , where h represents height, and a and b represent the two parallel sides. Then,

0 5 x + 1 d x = 1 2 h ( a + b ) = 1 2 · 5 · ( 1 + 6 ) = 35 2 .

Thus the average value of the function is

1 5 0 0 5 x + 1 d x = 1 5 · 35 2 = 7 2 .
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Find the average value of f ( x ) = 6 2 x over the interval [ 0 , 3 ] .


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Key concepts

  • 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
    a b f ( x ) d x = lim n i = 1 n f ( x i * ) Δ x
  • Properties of the Definite Integral
    a a f ( x ) d x = 0
    b a f ( x ) d x = a b f ( x ) d x
    a b [ f ( x ) + g ( x ) ] d x = a b f ( x ) d x + a b g ( x ) d x
    a b [ f ( x ) g ( x ) ] d x = a b f ( x ) d x a b g ( x ) d x
    a b c f ( x ) d x = c a b f ( x ) for constant c
    a b f ( x ) d x = a c f ( x ) d x + c b f ( x ) d x

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
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ramon Reply
Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
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J, combine like terms 7x-4y
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I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
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what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
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what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
can nanotechnology change the direction of the face of the world
Prasenjit Reply
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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Practice Key Terms 8

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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