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can you break this into \(4\) intervals \((4,6)\)?
I think I will just show you, and then it will make sense \(x_1=4, x_2=4.5, x_3=5,\) and \(x_4=6\)
So how did you choose those four numbers? I understand they are between 4 to 6 but is a random selection?
it would be the only way to break the interval \([4,6]\) into \(4\) parts.
Don't think so https://gyazo.com/f25d718a61e95059646b790eba350188
I meant \(x_4=5.5\) sorry
want me to explain why?
When we take the integral of something we are taking the area under the function, imagine you want to take the integral of the function \(f(x) = 2x+3\) from \(4\) to \(6\) The graph would look something like this|dw:1441154162616:dw|
we want the are under the graph from \(x=4, \) to \(x=6\). One way to approximate this area would be to break it up into four sections |dw:1441154252671:dw| Now we would estimate the area of each part We treat each part as a rectangle, so the area is 0.5 (the width of the rectangle) times \(f(4.5)\) . So the area of the first rectangle is estimated by \(0.5*f(4.5)\) and we get this area (approximately) |dw:1441154442521:dw|
If we break this into infinite pieces, we would get the exact area, not just the approximate.
I understand why you did so...just how? Like why .5 and not .3?
if I give you two dollars and ask you to break it into 4 even pieces, what would the size of each one be?
Ok, that's why, the pieces must be even. So why wasn't the 6 included? Is it because it's the last and should already be known?