Question

integral help

Answer 1

first or second?

Answer 2

both

Answer 3

Hello

Answer 4

the first one is a straight up calculator exercise
compute
\[f(1)+f(2)+f(3)+f(4)\]

Answer 5

I'm pretty sure that you can mange the very first problem on your own, right?

Answer 6

you are also supposed to multiply by \(\Delta x\) but since the distance between each of these numbers is 1, \(\Delta x=1\) so you can ignore that part

Answer 7

Definite integral is defined as following:\[\int\limits_{a}^{b}f(x)dx=\lim_{n \rightarrow \infty}\sum_{k=1}^{n}f(x _{k}) \Delta x\].

Answer 8

In which \[x _{k}=a+\frac{ b-a }{ n }k\]

Answer 9

Do you know what this means?

Answer 10

nope

Answer 11

You've heard of it, though, right?

Answer 12

ok.

Answer 13

do you know what does definite integral do?

Answer 14

you have to be honest. do you know it or not?

Answer 15

nope

Answer 16

I'll briefly inform you of what the heck definite integral does.

Answer 17

In short, it lets you find the "area" under a graph in an interval.

Answer 18

The reason I wrote "area" is because if the graph is below the x-axis, then the definite integral turns out to be negative.

Answer 19

But for this problem, the graph is over the x-axis in the interval [0,4], so we don't have to worry about the negative.

Answer 20

Here is a drawing that expresses what I said.

Answer 21

http://images.tutorvista.com/cms/images/113/definite-integral.png

Answer 22

Are you familiar with the definite integral notation?

Answer 23

k, i understand what you just said

Answer 24

a little bit

Answer 25

Good!
From what I said, if you understand the drawing from the link, then we're good to go to the next step.

Answer 26

kk

Answer 27

Great. Then, we'll move on to Riemann Sum.

Answer 28

A mathematician named Riemann (German) developed this method, called Riemann Sum, to calculate the definite integral.

Answer 29

okok

Answer 30

Do you remember the equation that I first gave you?

Answer 31

It looked like this: http://classconnection.s3.amazonaws.com/33333/flashcards/644875/jpg/area-sigma-notation-a.jpg

Answer 32

I just added that \[x _{i}=a+\frac{ b-a }{ n }i\]

Answer 33

You might get confused.
I first used k, then I used i.
They're basically the same, so you can ignore that little difference

Answer 34

but where does your RHS of your xi come from?

Answer 35

I'll explain it in a minute.

Answer 36

First, look at this drawing carefully.
http://sy48497.tripod.com/artifacts/webquest/maple/images/rsintro2.gif

Answer 37

Let's say that I want to find the area under curve of a random function from 0 to 10 using Riemann Sum.

Answer 38

The method of Riemann Sum is show in the drawing; you add up rectangles that are under the graph.

Answer 39

yeh k

Answer 40

For all of the rectangles, the width is the same.
We can say that the width of one rectangle is the change of x, or \[\Delta x\]

Answer 41

yep

Answer 42

For this particular graph, the definite integral started from 0, right?

Answer 43

Now, this is where some imagination and thinking come into play.
Assume that under the graph, there are 'n' rectangles, ok?

Answer 44

yes it did. yeh k.

Answer 45

Now if n gets very very very close to infinity, what would happen to its width?

Answer 46

become very small?

Answer 47

Correct!

Answer 48

And at the same time, if you think about it, the sum of the rectangles would be almost the same as the area under the curve, right?

Answer 49

yes

Answer 50

Now, it might get a little complicated, so watch carefully.

Answer 51

From a to b, there are 'n' rectangles, so we can say that the width of one rectangle is \[\frac{ b-a }{ n }\], which is \[\Delta x\]

Answer 52

And, xi can be expressed in terms of Delta x as following
\[x _{i}=a+\Delta x i=a+\frac{ b-a }{ n }i\]

Answer 53

Is there anything that you're confused on so far?

Answer 54

Hey

Answer 55

Can I ask you something?

Answer 56

For (b), is it asking to write the notation for definite integral, or actually calculate it?

Answer 57

If it is asking you to just write the notation, write\[\int\limits_{0}^{4}(2x+4)dx\] which means the "area" under the curve, y=2x+4, in the interval from 0 to 4

Answer 58

If it is asking you to actually calculate it, then watch these videos to apply the method to the problem; https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/riemann-sums/v/simple-riemann-approximation-using-rectangles

I'm truly sorry for not being able to help you, but I'm in Korea, which means that it's 11:50, so I have to sleep for school tomorrow.
Good luck!

Answer 59

k thanks anyway