DAILY LIFE USES U SES OF DIFFERENTIATION AND INTEGRATION In Isaac Newton's day da y, one of the biggest problems was poor navigation at sea. Before calculus was developed, the stars were vital for navigation. Shipwrecks occurred because the ship was not where the captain thought it should be. There was not a good enough understanding of how the arth, stars and planets moved with respect to each other. !alculus "differentiation and integration# was developed to improve t his understanding. $ifferentiation and integration can help us solve many types of real%world problems. &e use the derivative to determine det ermine the maimum and minimum values of particular functions "e.g. cost, strength, amount of material used in a building, profit, loss, etc.#. $erivatives are met in many engineering and science problems, especially when modeling the behavior of moving ob(ects. )ur discussion begins with some general applications which we can then apply to specific problems. *. It is used !)N)+I! a lot, calculus is also a base of economics .it is used in history, for predicting the life of a stone -.it is used in geography geograph y, which is used to study the gases present in the atmosphere . It is mainly used in daily by pilots to measure the pressure n the air. Shipwrecks occurred because the ship was not where the captain thought it should be. There was not a good enough understanding of how the arth, stars and planets moved with respect to each other. !alculus "differentiation and integration# was developed to improve t his understanding. $ifferentiation and integration can help us solve many types of real%world problems. &e use the derivative to determine det ermine the maimum and minimum values of particular functions "e.g. cost, strength, amount of material used in a building, profit, loss, etc.#. $erivatives are met in many engineering and science problems, especially when modeling the behavior of moving ob(ects. INTEGRATION/ *. 0pplications 0pplications of the Indefinite I ndefinite Integral shows how to find displacement "from velocity# and velocity "from acceleration# using the indefinite integral. There are also some electronics applications.
In primary school, we learnt how to find areas of shapes with straight sides "e.g. area of a triangle or rectangle#. But how do you find areas when the sides are curved1 e.g. . 0rea under a !urve and -. 0rea in between the two curves. 0nswer is by Integration. . 2olume 2olume of Solid of 3evolution eplains how to use integration to find the volume of an ob(ect with curved sides, e.g. wine barrels. 4. !entroid of an 0rea means the cenetr of mass. &e see how to use integration to find the centroid of an area with curved sides. 5. +oments of Inertia eplain how to find the resistance of a rotating body. &e use integration when the shape has curved sides. 6. &ork by a 2ariable 7orce shows how to find the work done on an ob(ect when the force is not constant. 8. lectric !harges have a force between them that varies depending on the amount of charge and the distance between the charges. &e use integration to calculate the work done when charges are separated. 9. 0verage 0verage 2alue 2alue of a curve can be calculated using integration.
In this section we will show how the definite integral can be used in different applications. Some of the concepts may sound new to the reader, but we will eplain what you need to comprehend as we go along. &e will take three applications/ The concepts of work from physics, fluid statics from engineering, and the normal probability from statistics.
Work &ork in physics is defined as the product of the force and displacement. 7orce and displacement are vector :uantities, which means they have a direction and a magnitude. 7or eample, we say the compressor eerts a force of upward. The magnitude here is and the direction is upward. ;owering a book from an upper shelf to a lower one by a distance of away from its initial position is another eample of the vector nature of the displacement.
where is the force and is the displacement. If the force is measured in Newtons and distance is in meters, then work is measured in the units of energy which is in (oules Example 1: >ou push an empty grocery cart with a force of
for a distance of
Solution: ?sing the formula above,
Example 2: 0 librarian librarian displaces a book from an upper upper shelf to a lower one. If the vertical distance between the two shelves is and the weight of the book is .
Thus
0 bucket bucket has an empty weight weight of . It is filled with sand of weight and attached to a rope of weight . Then it is lifted from the floor at a constant rate to a height above the floor. &hile in flight, the bucket leaks sand grains at a constant rate, and by the time it reaches the top no sand is left in the bucket. 7ind the work done/ *.
by lifting the empty bucket@
.
by lifting the sand alone@
-.
by lifting the rope alone@
.
by th the e lif lifti ting ng th the e buc bucke ket, t, th the e san sand, d, an and d the the ro rope pe to toge geth ther er..
Solution: *. The empty bucket. Since the bucketAs weight is constant, the worker must eert a force that is e:ual to the weight of the empty bucket. Thus
. The sand alone. The weight of the sand is decreasing at a constant rate from to over the lift. &hen the bucket is at above the floor, the sand weighs
The graph of represents the variation of the force with height "7igure -#. The work done corresponds to computing the area under the force graph.
Thus the work done is
-. The rope alone. Since the weight of the rope is and the height is , the total weight of the rope from the floor to a height of
is
But since the worker is constantly pulling the rope, the ropeAs length is decreasing at a constant rate and thus its weight is also decreasing as the bucket being lifted. So at , the there remain to be lifted of weight . Thus the work done to lift the weight of the rope is
. The bucket, the sand, and the rope together .
Fluid Sai!": #re""ure >ou have probably studied that pressure that pressure is is defined as the force per area
which has the units of ascals or Newtons per meter s:uared, In the study of fluids, such as water pressure on a dam or water pressure in the ocean at a depth another e:uivalent formula can be used. It is called the liquid pressure at depth /
where is the weight density , which is the weight of the column of water per unit volume. 7or eample, if you are diving in a pool, the pressure of the water on your body can be measured by calculating the total weight that the column of water is eerting on you times your depth. 0nother way to epress this formula, the weight density is defined as
where
is the density of the fluid and
is
is the acceleration due to gravity "which
on arth#. The pressure then can be written as
Example $: &hat is the total pressure eperienced by a diver in a swimming pool at a depth of 1
Solution 7irst we calculate the fluid pressure the water eerts on the diver at a depth of /
The density of water is
, thus
The total pressure on the diver is the pressure due to the water plus the atmospheric pressure. If we assume that the diver is located at sea%level, then the atmospheric pressure at sea level is about . Thus the total pressure on the diver is
Example %: &hat is the fluid pressure "ecluding the air pressure# and force on the top of a flat circular plate of radius that is submerged horiContally at a depth of 1 Solution// Solution The density of water is
Since the force is
. Then
then
0s you can see, it is easy to calculate the fluid force on a horiContal horiContal surface because because each point on the surface is at the same depth. The problem becomes a little l ittle complicated when we want to calculate the fluid force or pressure if the surface is vertical. In this situation, the pressure is not constant at every point because the depth is not constant at each point. To To find the fluid force or pressure on a vertical surface we must use calculus. T&e Fluid For!e o' a (eri!al Sur)a!e Suppose a flat surface is submerged vertically in a fluid of weight density w and the submerged portion of the surface etends from to along the vertical ais, whose positive direction is taken as downward. If is the width of the surface and is the depth of point then the fluid force is defined as
Example *: 0 perfect perfect eample of a vertical surface is the face of a dam. &e can picture it as a rectangle of a certain height and certain width. ;et the height of the dam be and of width of . 7ind the total fluid force eerted on the face if the top of the dam is level with the water surface "7igure #.
Solution: ;et the depth of the water. 0t an arbitrary point on the dam, the width of the dam is and the depth is . The weight density of water is
?sing the fluid force formula above,
Normal #ro+a+iliie" If you were told by the postal service that you will receive the package that you have been waiting for sometime tomorrow, what is the probability that you will receive it sometime between -/== + and 4/== + if you know that the postal serviceAs hours of operations are between 6/== 0+ to 5/== +1 If the hours of operations are between 6 0+ to 5 +, this means they operate for a total of . The interval between - + and 4 + is , and thus the probability that your package will arrive is
So there is a probability of that the postal service will deliver your package sometime between the hours of - + and 4 + "or during any interval#. That is easy enough.
where is called the the probability probability density function "pdf#. function "pdf#. )ne of the most useful probability density functions is the normal curve or curve or the Gaussian curve "and curve "and sometimes the bell curve# curve# "7igure 5#. This function enables us to describe an entire population based on statistical measurements taken from a small sample of the population. The only measurements needed are the mean and the standard deviation . )nce those two numbers are known, we can easily find the normal curve by using the following formula.
T&e Normal #ro+a+ili, De'"i, Fu'!io' The Daussian curve for a population with mean by
and standard deviation
is given
where the factor is called the normalization constant. It is needed to make the probability over the entire space e:ual to That is,
Example -: Suppose that boes containing each and a standard deviation of *.
tea bags have a mean weight of
&hat &h at pe perc rcen enta tage ge of al alll th the e bo boe es s is e epe pect cted ed to we weig igh h be betw twee een n 1
.
&ha &h at is th the e pr prob obab abil ilit ity y th tha at a bo bo wei eigh ghs s le less ss th than an
1
-.
&ha &h at is th the e pr prob obab abil ilit ity y th tha at a bo bo wi will ll wei eigh gh e eac actl tly y
1
Solution: *. ?sing the normal probability density function,
and an d
Substituting for
and
we get
The percentage of all the tea boes that are epected to weight between ounces can be calculated as
and
The integral of does not have an elementary anti%derivative and therefore cannot be evaluated by standard techni:ues.
That is,
Te!&'olo., Noe: To Noe: To make this computation with a graphing calculator of the TI% 8-E8 family, do the following/ •
7rom the /DISTR0 /DISTR0 menu menu "7igure 6# select option , which puts the phrase FnormalcdfF in the home screen. 0dd lower bound, upper bound, mean, standard deviation, separated by commas, close the parentheses, and press /ENTER0 /ENTER0.. The result is shown in 7igure 8.
Fi.ure 2-
Fi.ure 2 . 7or the probability that a bo weighs less than under the curve to the left of Since the value of than a billionth#,
getting the area between numerically, we get
and
which says that we would epect
, we use the area is very small "less
will yield a fairly good answer. Integrating
of the boes to weigh less than
-. Theoretically the probability here will be eactly Cero because we will be integrating from to which is Cero.
&hat &h at is th the e pe perc rcen enta tage ge of th the e po popu pula lati tion on th that at ha has s a sc scor ore e be betw twee een n
.
&ha &h at pe perc rcen enta tage ge of th the e po popu pula lati tion on ha has s a sc scor ore e ab abov ove e
Solution:
1
and an d
1
*. ?sing the normal probability density function,
and substituting
and
The percentage of the population that has a score between
and
is
0gain, the integral of does not have an elementary elementary anti%derivative and therefore therefore cannot be evaluated. ?sing the programing feature of a scientific calculator or a mathematical computer software, we get
That is,
&hich says that
of the population has an IG score between
and
. To measure the probability that a person selected randomly will have an IG score above ,
This integral is even more difficult to integrate since it is i s an improper integral. To avoid the messy work, we can argue that since it is etremely rare to meet someone with wi th an IG sc scor ore e of ov over er we ca can n ap appr pro oim imat ate e th the e in inte tegr gral al fr from om to Then Th en
Integrating numerically, we get
So the probability of selecting at random a person with an IG score above is . ThatAs about one person in every individualsH
