Question

Find (click link for image: http://i.imgur.com/UQDs9.gif ) by evaluating an appropriate definite integral over the interval [0,1].

Answer 1

fix: sin should be in the numerator

Answer 2

\[\large\lim_{n\to\infty}\sum_{i=1}^n\frac{\sin(\frac{i\pi} n)}n\]

Answer 3

I don't understand why you are integrating over [0,1] and not [1,+infty]
are you sure you typed the right interval?

Answer 4

oh wait... I think I might see what they want...

Answer 5

Yes, I typed it correctly. If it helps I'm in the u-substitution section

Answer 6

the integral over the interval [a,b] of a function f(x) is given by \[\large\lim_{n\to\infty}\sum_{i=1}^nf(a+i\Delta x)\Delta x~;~~\Delta x=\frac{b-a}n\]you have been given\[\large\lim_{n\to\infty}\sum_{i=1}^n\frac{\sin(\frac{i\pi} n)}n\]so it looks like in your case\[\Delta x=\frac1n\text{ and }f(x)=\sin(\pi x)\]

Answer 7

so in your case have been given\[\large\lim_{n\to\infty}\sum_{i=1}^n\frac{\sin(\frac{i\pi} n)}n=\int_0^1\sin(\pi x)dx\]which can be done with the u-sub\[u=\pi x\]

Answer 8

where does the \[\Delta x\] come in

Answer 9

\[\lim_{n\to\infty}\Delta x=\lim_{n\to\infty}\frac{b-a}n=dx\]...sort of
or do you mean where is it in the summand?

Answer 10

I guess I'm just generally confused, sums were not my strong point

Answer 11

you don't actually have to to any summations here, just notice the constituent parts and identify what integral ti represents

Answer 12

it*

Answer 13

your interval is \([a,b]=[0,1]\) so we have\[\large\lim_{n\to\infty}\sum_{i=1}^nf(a+i\Delta x)\Delta x~;~~\Delta x=\frac{b-a}n\]for you\[\Delta x=\frac1n=\frac{1-0}n\]you have been given\[\large\lim_{n\to\infty}\sum_{i=1}^n\frac{\sin(\frac{i\pi} n)}n=\large\lim_{n\to\infty}\sum_{i=1}^n\sin[a+\pi i(\frac1n)]\frac1n\]since \(a=0\) this becomes\[\large\lim_{n\to\infty}\sum_{i=1}^n\sin[\pi i(\frac1n)]\frac1n=\lim_{n\to\infty}\sum_{i=1}^n\sin[\pi i\Delta x]\Delta x\]try to compare the two with the definition of the integral and you see that\[\large\lim_{n\to\infty}\sum_{i=1}^n\sin[\pi i\Delta x]\Delta x=\int_0^1\sin(\pi x)dx\]

Answer 14

so from there I just set u = pi*x ... ?

Answer 15

yep

Answer 16

So then ... du = pi, but then \[\int\limits_{1}^{0}\sin \pi(u)\]
but sin pi is 0 ... I'm probably being an idiot with calculations right now so

Answer 17

Lol switch the interval

Answer 18

1)\[u=\pi x\implies du=\pi dx\implies dx=\frac{du}\pi\](don't forget dx, it is important!)
2) change the bounds\[u=\pi x\implies x=0\to u=0\text{ and }x=1\to u=\pi\]
which leads to\[\int_0^1\sin(\pi x)dx=\int_0^\pi\sin(u)(\frac{du}\pi)=\frac1\pi\int_0^\pi\sin udu\]
3) you haven't integrated it yet. you won't get zero after integrating

Answer 19

Okay thanks so much

Answer 20

welcome!

Answer 21

Luella has a pond in her yard. She wants to surround it with 330 feet of fencing. The fencing is sold by the yard. How many yards should she buy?

Answer 22

someone check my answers
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Answer 23

https://i.imgur.com/Wjmpc6D.png

Answer 24

Assessment items
Assessment started: undefined.
Item 1

Read the passage.

Dark Star

Giselle leaned against the mahogany bay horse, stroking the white diam

Answer 25

Okay I was gone for a minute BUT I'M BACK NOW

Answer 26

the scatter plot presents Ray's monthly electricity bills and amount of time ray

Answer 27

Which word shares a connotative meaning with the word cheap as it is used in this sentence?

Ashley thought her parents were being cheap when they bought ge

Answer 28

Which revision fixes the dangling modifier in the sentence?

Rowing through the rapids, the canoe began to fill with water.

Answer 29

dangling modifier

Answer 30

Misplaced Modifiers

Answer 31

For which literary work did Joseph Heller use his military experiences as a prima

Answer 32

Calc III Tutorial: Lagrange Multipliers

Answer 33

Can someone translate this for me?

Answer 34

Find the interest due on $1,090 at 8% for 5 years.

43.60
68.13
436.00
681.25

Answer 35

Now identify the simple subject and simple predicate.

The meeting began promptly at seven o'clock.

Answer 36

What can be inferred from these lines describing the pardoner in Chaucer's prolog

Answer 37

What is the rate of change of the linear relationship modeled in the table?

x y

Answer 38

Which of these best identifies the shift of mood in this sentence?

"If you were to sit more comfortably, you will not fall off.

Answer 39

Which scenario best matches the linear relationship expressed in the equation y = u221214x + 1,700?

Kent has $1,700 in his bank account and spends $14 each

Answer 40

The saying "Whoso is a man must be a nonconformist" is based on whose transcendentalist belief?
A.