00:01
Alright, so in this question, we're given a function f of x equals 1 over x, along with 4x values, x sub 1 through x sub 4.
00:08
And as you can see, the size or the step size between these x values is 1⁄af, so our delta x is 1⁄2.
00:16
In part a, this question, we're asked to calculate the sum i equals 1 to 4 of f of x sub i times delta x.
00:27
So before i plug anything in, i'm just going to write this sum out, expand it, so it's easier to see what we're working with.
00:34
When i equals 1, i would have f of x sub 1 times delta x, then i is 2, so we have f of x sub 2 times delta x.
00:48
I is 3, we have f of x sub 3 times delta x.
00:55
And finally, i is 4, so we have f of x sub 4 times delta x.
01:01
Now we can start plugging things in, so first i have that i need to find f of x of 1.
01:07
X of 1 is 1 half.
01:09
F of x is 1 over x, so 1 over 1 half gives me 2, and i know that delta x is 1 half, which was given in the problem.
01:21
F of x of 2, now moving on, would be 1 over 1, which gives me 1 times that same delta x of 1.
01:28
For f of x of three, i have 1 over 3 halves, which would be 2 thirds, times 1 half.
01:37
And finally, f of x of 4, x of 4 is 2, so i get 1 over 2 times 1 half for delta x.
01:46
So looking at these, i have 1 plus 1 1 1 1 3 plus 1 quarter.
01:55
I went ahead and created a common denominator to add these fractions together.
01:58
I'm not going to write that out, but that's not too bad.
02:01
And we're going to end up with a sum of 2512s, which is the answer to a.
02:07
So this is actually an estimation of the area under the curve f of x or the area under 1 over x between some x values.
02:20
So we are essentially estimating a definite integral.
02:24
In part b, we're asked to write out this definite integral that we estimated.
02:29
So to do that, i'm going to draw out kind of the situation so we can see visually what we're looking at.
02:39
So i'm just going to leave all my axes.
02:41
This is just going to be a really rough sketch...