What is the integral of sin(x) dx from 0 to 2pi?

[ad_1]

An alternative way to do this starting from the limit definition is:

Don't use plagiarized sources. Get Your Custom Essay on
What is the integral of sin(x) dx from 0 to 2pi?
Just from $13/Page
Order Essay

##int_(a)^(b) f(x)dx = lim_(n->oo) sum_(i=1)^N f(x_i^”*”)Deltax##

where:

  • ##n## is the number of rectangles used to approximate the integral, i.e. the area between the curve and the x-axis.
  • ##i## is the index of each rectangle in ##[0,2pi]##.
  • ##N## is the index of the final rectangle in ##[0,2pi]##.
  • ##f(x_i^”*”)## is the height of each given rectangle in ##[0,2pi]##, which varies as ##sin(x)##.
  • ##Deltax## is the width of each given rectangle in ##[0,2pi]##, which converges to ##0## as ##n->oo##.

If we use the midpoint-rectangular approximation method (MRAM), we choose a convenient interval ##Deltax## such that we can find a midpoint for each rectangle of dimension ##Deltax xx f(x_i^”*”)##, where the midpoint of the ##i##th rectangle is defined as

##M_i = x_(i-1)+(x_i – x_(i-1))/2##.

Let us choose ##Deltax = pi/2## such that ##x = {0,pi/2,pi,(3pi)/2,2pi}## for ##[0,2pi]## and ##n = {1,2,3,4}##.

Then each rectangle’s width is ##pi/2##, and:

Each corresponds to an ##f(x_i^”*”)## that gives you the height of the ##i##th rectangle as

##f(x_1^”*”) ~~ f(M_1) = sin(x_0+(x_1-x_0)/2) = sin(pi/4) = sqrt2/2,##

##f(x_2^”*”) ~~ f(M_2) = sin(x_1+(x_2-x_1)/2) = sin((3pi)/4) = sqrt2/2,##

##f(x_3^”*”) ~~ f(M_3) = sin(x_2+(x_3-x_2)/2) = sin((5pi)/4) = -sqrt2/2,##

##f(x_4^”*”) ~~ f(M_4) = sin(x_3+(x_4-x_3)/2) = sin((7pi)/4) = -sqrt2/2.##

In the end, what you get from MRAM is the following result:

##color(blue)(int_(0)^(2pi) sin(x)dx ~~ lim_(n->4) sum_(i=1)^4 sin(M_i)Deltax)##

##= (sin(pi/4) + sin((3pi)/4) + sin((5pi)/4) + sin((7pi)/4))*Deltax##

##= (sqrt2/2 + sqrt2/2 – sqrt2/2 – sqrt2/2) * pi/2##

##= color(blue)(0)##

Which is not surprising given that ##sin(x)## in ##[0,pi]## is equal to ##-sin(x)## in ##[pi,2pi]##, which means that

##int_(0)^(pi) sinxdx = -int_(pi)^(2pi) sinxdx##,

and thus:

##color(green)(int_(0)^(2pi) sinxdx)##

##= int_(0)^(pi) sinxdx + int_(pi)^(2pi) sinxdx##

##= color(green)(0)##,

regardless of the chosen method.

Note that RRAM, LRAM, and MRAM are approximations, so it was a coincidence that it gave the exact answer.



[ad_2]

Source link

Place your order
(550 words)

Approximate price: $22

Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2018 at 10:52 AM
Total price:
$26
The price is based on these factors:
Academic level
Number of pages
Urgency
Basic features
  • Free title page and bibliography
  • Unlimited revisions
  • Plagiarism-free guarantee
  • Money-back guarantee
  • 24/7 support
On-demand options
  • Writer’s samples
  • Part-by-part delivery
  • Overnight delivery
  • Copies of used sources
  • Expert Proofreading
Paper format
  • 275 words per page
  • 12 pt Arial/Times New Roman
  • Double line spacing
  • Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Our guarantees

Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read more

Zero-plagiarism guarantee

Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read more

Free-revision policy

Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read more

Privacy policy

Your email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read more

Fair-cooperation guarantee

By sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more

Order your essay today and save 15% with the discount code BANANA