<< Chapter < Page Chapter >> Page >

Express cos x d x as an infinite series. Evaluate 0 1 cos x d x to within an error of 0.01 .

C + n = 1 ( −1 ) n + 1 x n n ( 2 n 2 ) ! The definite integral is approximately 0.514 to within an error of 0.01 .

Got questions? Get instant answers now!

As mentioned above, the integral e x 2 d x arises often in probability theory. Specifically, it is used when studying data sets that are normally distributed, meaning the data values lie under a bell-shaped curve. For example, if a set of data values is normally distributed with mean μ and standard deviation σ , then the probability that a randomly chosen value lies between x = a and x = b is given by

1 σ 2 π a b e ( x μ ) 2 / ( 2 σ 2 ) d x .

(See [link] .)

This graph is the normal distribution. It is a bell-shaped curve with the highest point above mu on the x-axis. Also, there is a shaded area under the curve above the x-axis. The shaded area is bounded by alpha on the left and b on the right.
If data values are normally distributed with mean μ and standard deviation σ , the probability that a randomly selected data value is between a and b is the area under the curve y = 1 σ 2 π e ( x μ ) 2 / ( 2 σ 2 ) between x = a and x = b .

To simplify this integral, we typically let z = x μ σ . This quantity z is known as the z score of a data value. With this simplification, integral [link] becomes

1 2 π ( a μ ) / σ ( b μ ) / σ e z 2 / 2 d z .

In [link] , we show how we can use this integral in calculating probabilities.

Using maclaurin series to approximate a probability

Suppose a set of standardized test scores are normally distributed with mean μ = 100 and standard deviation σ = 50 . Use [link] and the first six terms in the Maclaurin series for e x 2 / 2 to approximate the probability that a randomly selected test score is between x = 100 and x = 200 . Use the alternating series test to determine how accurate your approximation is.

Since μ = 100 , σ = 50 , and we are trying to determine the area under the curve from a = 100 to b = 200 , integral [link] becomes

1 2 π 0 2 e z 2 / 2 d z .

The Maclaurin series for e x 2 / 2 is given by

e x 2 / 2 = n = 0 ( x 2 2 ) n n ! = 1 x 2 2 1 · 1 ! + x 4 2 2 · 2 ! x 6 2 3 · 3 ! + + ( −1 ) n x 2 n 2 n · n ! + = n = 0 ( −1 ) n x 2 n 2 n · n ! .

Therefore,

1 2 π e z 2 / 2 d z = 1 2 π ( 1 z 2 2 1 · 1 ! + z 4 2 2 · 2 ! z 6 2 3 · 3 ! + + ( −1 ) n z 2 n 2 n · n ! + ) d z = 1 2 π ( C + z z 3 3 · 2 1 · 1 ! + z 5 5 · 2 2 · 2 ! z 7 7 · 2 3 · 3 ! + + ( −1 ) n z 2 n + 1 ( 2 n + 1 ) 2 n · n ! + ) 1 2 π 0 2 e z 2 / 2 d z = 1 2 π ( 2 8 6 + 32 40 128 336 + 512 3456 2 11 11 · 2 5 · 5 ! + ) .

Using the first five terms, we estimate that the probability is approximately 0.4922 . By the alternating series test, we see that this estimate is accurate to within

1 2 π 2 13 13 · 2 6 · 6 ! 0.00546 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use the first five terms of the Maclaurin series for e x 2 / 2 to estimate the probability that a randomly selected test score is between 100 and 150 . Use the alternating series test to determine the accuracy of this estimate.

The estimate is approximately 0.3414 . This estimate is accurate to within 0.0000094 .

Got questions? Get instant answers now!

Another application in which a nonelementary integral arises involves the period of a pendulum. The integral is

0 π / 2 d θ 1 k 2 sin 2 θ .

An integral of this form is known as an elliptic integral of the first kind. Elliptic integrals originally arose when trying to calculate the arc length of an ellipse. We now show how to use power series to approximate this integral.

Period of a pendulum

The period of a pendulum is the time it takes for a pendulum to make one complete back-and-forth swing. For a pendulum with length L that makes a maximum angle θ max with the vertical, its period T is given by

T = 4 L g 0 π / 2 d θ 1 k 2 sin 2 θ

where g is the acceleration due to gravity and k = sin ( θ max 2 ) (see [link] ). (We note that this formula for the period arises from a non-linearized model of a pendulum. In some cases, for simplification, a linearized model is used and sin θ is approximated by θ . ) Use the binomial series

1 1 + x = 1 + n = 1 ( −1 ) n n ! 1 · 3 · 5 ( 2 n 1 ) 2 n x n

to estimate the period of this pendulum. Specifically, approximate the period of the pendulum if

  1. you use only the first term in the binomial series, and
  2. you use the first two terms in the binomial series.
    This figure is a pendulum. There are three positions of the pendulum shown. When the pendulum is to the far left, it is labeled negative theta max. When the pendulum is in the middle and vertical, it is labeled equilibrium position. When the pendulum is to the far right it is labeled theta max. Also, theta is the angle from equilibrium to the far right position. The length of the pendulum is labeled L.
    This pendulum has length L and makes a maximum angle θ max with the vertical.

We use the binomial series, replacing x with k 2 sin 2 θ . Then we can write the period as

T = 4 L g 0 π / 2 ( 1 + 1 2 k 2 sin 2 θ + 1 · 3 2 ! 2 2 k 4 sin 4 θ + ) d θ .
  1. Using just the first term in the integrand, the first-order estimate is
    T 4 L g 0 π / 2 d θ = 2 π L g .

    If θ max is small, then k = sin ( θ max 2 ) is small. We claim that when k is small, this is a good estimate. To justify this claim, consider
    0 π / 2 ( 1 + 1 2 k 2 sin 2 θ + 1 · 3 2 ! 2 2 k 4 sin 4 θ + ) d θ .

    Since | sin x | 1 , this integral is bounded by
    0 π / 2 ( 1 2 k 2 + 1.3 2 ! 2 2 k 4 + ) d θ < π 2 ( 1 2 k 2 + 1 · 3 2 ! 2 2 k 4 + ) .

    Furthermore, it can be shown that each coefficient on the right-hand side is less than 1 and, therefore, that this expression is bounded by
    π k 2 2 ( 1 + k 2 + k 4 + ) = π k 2 2 · 1 1 k 2 ,

    which is small for k small.
  2. For larger values of θ max , we can approximate T by using more terms in the integrand. By using the first two terms in the integral, we arrive at the estimate
    T 4 L g 0 π / 2 ( 1 + 1 2 k 2 sin 2 θ ) d θ = 2 π L g ( 1 + k 2 4 ) .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
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?
Abdul Reply
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?
Abdul
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 2' conversation and receive update notifications?

Ask