WEBVTT
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Yeah. So problem number 10. We have a
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function F of X. Cosine of X on the
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interval from zero to pi over two and they want
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us to approximate the area. Um sampling it with
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the computer using 10, 30 50 and 100 rectangles
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. So real rough, you know that when you
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graph the co sign function from zero two, pi
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over to co sign function starts at the value of
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one and it's gonna end at zero. So what
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you see this curve doing is something like this is
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an important thing to note is this curve lies entirely
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above the X axis. So when I create the
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rectangles, they're all going to be a positive height
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and I add up the area of all the rectangles
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, they're all positive. That will approximate this area
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. So I'm going to use my computer to do
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this. That the key is. What is the
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interval, the length, the width of this interval
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here from here to here is Pi over two.
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Okay, so when I divide that by 10 so
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if we just take a look real quick, what
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are going to be the widths of our rectangles?
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Okay When I have 10 rectangles, the width of
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each rectangle Is Pi over 20. When I have
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30 rectangles, the width of each rectangle is pi
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over 60. When I have 50 rectangles, each
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rectangle is pi over 100. And at 100 rectangles
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each rectangle is pi over 200 in width. Okay
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so I'm gonna have 10 rectangles of with pi over
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20. That will cover the interval from zero to
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pi or two. So we're going to use the
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computer to make all of this happen. So um
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let's get our calculator for all of this and let's
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just figure out so the function was co sign of
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X. And the co sign of X. And
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I'm going to store that. That is my function
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F. Of X. I know that now F
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. Of zero should be one. Okay that's what
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the coastline function does. Now let's just create a
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sequence of numbers. And so I'm going to start
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at so in that first case remember I start at
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Pi over 20 if I'm using the right side.
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So pi over 20 is where I start. So
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let's just do that real quick. That's going to
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be sorry about that exit out of that. Um
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It's going to be pie over 20 is the first
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right point. And then I'm going from there to
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hire her to. Yeah shit. Yeah And I'm
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doing in steps of pi over 20 So pi over
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20. Yeah is how I'm stepping and I want
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to store all of that. I'm going to call
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those my X. Values. So that should create
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for me a list of 10 Right in points of
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length. Power over 20. So remember the width
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is what Pi over 20 in this case the pie
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over 20 is the width. So now to figure
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out the area it's going to be So pi divided
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by 20 times the sum of all of those X
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. Values. So F. The X. Values
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. Yeah. 0919403. So that 1st 1919403.
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Let's write that down. So 919403. That's my
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approximation of the area. When I have 10 rectangles
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. Now let's just go back and it should be
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a little bit quicker now when I do this with
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30 rectangles. So now the width is Pi over
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60. So I'm going to go over here and
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go back to the sequencing And I'm going to say
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now the width is Pi over 60. Yeah I'm
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going to pi over two and I'm going to increment
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and pi over 60. Those are my new X
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. Values. and now the width of each rectangle
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is Pi over 60. So I need to figure
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out what is pie divided by 60 times the sum
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of all of those X. Values. 973592 973592
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. Mhm. Yeah. Yeah. Yeah. Okay
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let's go up to 50. So 50 the width
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is going to be pi over 100. So let's
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just go back to our calculator and we'll just redo
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that sequence command. Now what changes is the width
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is pi over 100. Mhm. So that gives
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me 100 values each of with pi over 100.
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And now I'll just come in and The width is
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Pi over 100, Sum up all of those values
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. 984-1 is the approximation here, so.984-1.
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Yeah. Yeah. Mhm. Yeah. And now
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let's go to 100 rectangles. The width is Pi
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over 200. So go back to my calculator,
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let's just recall all of that command with all the
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typing And we're going to change this the pi over
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200 as the width and the implement. And then
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go back to where we did that some. And
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now just change this from pi over 100 pirates 200
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99215 so.99215 mm. Now they're saying guess where
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this is headed. I don't know exactly. That's
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why it's a guess these numbers look like they're getting
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closer and closer to one. So my guess is
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my guess is that the exact area? Yeah.
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Yeah. Is one square unit. Okay. So
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I don't know that, but just by trending by
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using more and more direct angles, that's my guess
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is that is headed to one.