Question

List x1, x2, x3, x4 where xi is the left endpoint of the four equal intervals used to estimate the area under the curve of f(x) between x = 4 and x = 6.

Answer 1

can you break this into \(4\) intervals \((4,6)\)?

Answer 2

I'm not understanding what it's looking for here.....
@zzr0ck3r How would I do that? I don't recall anything like that.

Answer 3

I think I will just show you, and then it will make sense

\(x_1=4, x_2=4.5, x_3=5,\) and \(x_4=6\)

Answer 4

So how did you choose those four numbers? I understand they are between 4 to 6 but is a random selection?

Answer 5

it would be the only way to break the interval \([4,6]\) into \(4\) parts.

Answer 6

Don't think so
https://gyazo.com/f25d718a61e95059646b790eba350188

Answer 7

I meant \(x_4=5.5\) sorry

Answer 8

want me to explain why?

Answer 9

Please

Answer 10

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

Answer 11

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



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)

Answer 12

If we break this into infinite pieces, we would get the exact area, not just the approximate.

Answer 13

I understand why you did so...just how? Like why .5 and not .3?

Answer 14

if I give you two dollars and ask you to break it into 4 even pieces, what would the size of each one be?

Answer 15

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?