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Welcome back to numerod.
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My name is kevin sharak.
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Let's take a look at this function here.
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So we have f of x is equal to x plus 6x squared.
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So x plus 6x squared.
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And we're looking at finding the area underneath this between 1 and 4.
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So x has to be greater than equal to 1 and less than or equal to 4.
00:28
All right.
00:29
So let's set up our summation.
00:30
So we're going to do k equals 1 up to n.
00:34
And this again is going to be f of 1 plus and it'll be 3 divided by n times k.
00:43
So 3 divided by n times k.
00:45
And that 3 comes from the difference between 4 and 1.
00:48
The divided by n gives us the width of each of those rectangles.
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And the k is the steps for each one.
00:52
And that's again going to be times 3 divided by n.
00:56
Because again, that's the width of each one of these rectangles.
00:59
So let's consider what happens with this.
01:01
We're going to do the summation again, k equals 1 to n, and then plug this in where we can.
01:08
So we're going to plug this in.
01:09
So this will be 1 plus 3k divided by n plus 6 1 plus 3k over n squared.
01:25
That'll be times 3 over n.
01:32
Let's continue simplifying.
01:33
So it's going to be the summation still.
01:38
We can simplify this up a little bit.
01:40
This will be 1 plus 3k over n plus, let's see, 6 plus 36, k6.
01:53
Squared over n squared plus 54 yeah it looks like 54 k squared over n squared oh 54 k squared over n squared that'll then be times 3 over n get ourselves a little bit more space here it's important you carry through that summation notation so that each line of work is still truthful so now we have 3 over n plus 3k over n squared plus 6 plus 36k over n squared plus 54 k squared oh i forgot that you're distributing you through plus tracking my own work there let's adjust all of these coefficients three times three would be nine there we go 9 k squared plus 18 uh three times 36 would be oh what would that be uh 108 108 108k, and then 54 times 3, what would 54 times 3, 54, 162.
03:23
My brain is a little tired, so i'm using my cat cutter here.
03:25
So it'll be 162 k squared over n cubed.
03:35
Very nice.
03:35
Okay, so now we can be, we can apply this summation to each one of these terms and begin to drag everything out.
03:42
This 3 divided by n is going to turn into just simply 3.
03:49
This 9k over n squared is going to be 9 over n squared times whatever k turns into after you do a summation for it...