00:01
So in this question, we are given an integral, which is the integral of x squared minus 2 between 0 and 2, which is back to x.
00:09
And we are asked to do four things with it.
00:11
We're firstly asked to sketch the bit inside the integral, the integral and the integral, which is x squared minus 2, on the interval, which is between 0 and 2.
00:20
We're then asked to calculate delta x and the subsequent set of points, x, x, is, from, from.
00:30
These.
00:32
The third thing is to calculate reamon sums.
00:35
So that's the left and the right riemann sums, given that n is 4 in the question.
00:43
And finally we're asked to determine which of our sums that we've calculated which one overestimates and which one underestimates the value of the integral.
00:55
So firstly we can start off by sketching the graph.
00:58
So the easiest way to do this is by plotting a couple of points if you don't know what it should look like.
01:02
So if we put in x equals 0, then we get minus 2.
01:06
If we put in x equals 1, then we'll get minus 1.
01:11
And if you put in x equals 2, then we'll get 2.
01:14
And you should know that an x squared parabola will look something like that roughly.
01:19
And so we can draw that side of it in, roughly that is only a sketch, so it doesn't need to be too accurate.
01:27
We then look across to the right here and we can start to calculate delta x.
01:32
So the formula for delta x is already written there is b minus a over n.
01:37
B and a we get from the question, so that's what we're integrating between.
01:41
So we integrate from a to b.
01:44
So b is 2 minus a which is 0 over n which is 4.
01:50
So that's 2 over 4 which is a half.
01:52
So delta x is a half.
01:55
Then change colour and we can work out our set of points xi which are given by the formula a plus i delta x for i between 0.
02:03
Now we know a from the question, we know our delta x because that's come from here.
02:11
And n we've been given up here as well.
02:14
So since n is 4, that means we'll have n plus 1 points.
02:19
So we'll have 5 points in total, x, nx1, x2, x3 and x4.
02:24
So using that formula, we see that x0 is going to be minus, or no, sorry, is going to be 0.
02:34
X1 is going to be a half, x2 is going to be 1, x3 is going to be 3 over 2, and x4 is going to be 2.
02:44
And if you want to, you can either use a formula or you can look at the graph.
02:48
So if we look at the graph, we see that our full region is this area between 0 and 2.
02:53
And with n is 4, so we're splitting it into four regions.
02:56
So if you split that equally up, you'll see you get a region between 0 and 1, 1 and 1 and 1 1 and 1.
03:03
And one and a half and two.
03:06
So that's that part of the question done.
03:09
Next we need to calculate our riemann sums.
03:12
Swap colour again for these.
03:14
So again i've got the formula for ream and sum up here, which is delta x times the sum of f of xi for i between zero and n, where the set of x i is given below, depending if you're doing the left or the right sum.
03:27
So for our left sum we've already calculated what our possible xi values could be up here but if you look at the intervals that we looked up with the graph we want the left -hand points of each of these intervals so that's that one is there for this interval it's that point there for this interval it's that point there and for this interval it's that point there and so those are the first four values we have up here so our set of xi values are going to be zero a half 1 and 3 of a 2.
04:02
So the sum is going to be equal to delta x, which we calculated up here was a half.
04:07
So a half times the sum of...
04:11
So before we do that, we need to work out our f of xis.
04:14
Sorry, so our set of f of xis are got by just plugging those xi values into the formula we have up here, x squared minus 2.
04:25
So 0 minus 2 is minus 2.
04:27
Half squared is a quarter, minus 2, it's minus 7 over 1.
04:31
4.
04:32
1 squared is 1 minus 2 is minus 1 and 3 over 2 squared is 9 over 4 minus 2 is 1 over 4 or a quarter...