WEBVTT
00:00:00.124 --> 00:00:13.654
Given π of π₯ is equal to π₯ squared minus four and π₯ is greater than or equal to negative four and less than or equal to two, evaluate the Riemann sum for π with six subintervals, taking sample points to be midpoints.
00:00:13.984 --> 00:00:24.314
Remember, we can approximate the definite integral of a function between the limits π and π by splitting the area between the curve and the π₯-axis into π subintervals.
00:00:24.654 --> 00:00:31.694
The width of each rectangle is given by π₯π₯, which is found by subtracting π from π and dividing this value by π.
00:00:32.084 --> 00:00:35.804
In this example, we want to split the area into six subintervals.
00:00:35.974 --> 00:00:40.384
So we let π be equal to six, and then π is negative four and π is two.
00:00:40.704 --> 00:00:47.404
The width of each of our subintervals is given then by two minus negative four over six, which is equal to one.
00:00:47.824 --> 00:00:53.914
Next, weβre going to sketch the curve of π¦ equals π₯ squared minus four between the endpoints negative four and two.
00:00:54.184 --> 00:00:56.714
And weβre going to split this into six subintervals.
00:00:56.964 --> 00:00:59.274
This question requires us to use midpoints.
00:00:59.494 --> 00:01:04.854
So the height of each rectangle will be equal to the value of the function at the middle of each subinterval.
00:01:05.034 --> 00:01:07.484
Thatβs going to look a little something like this.
00:01:07.684 --> 00:01:13.584
Now, we can work out the π₯-value at the end of each rectangle by repeatedly adding one to four.
00:01:13.894 --> 00:01:15.784
And that gives us each of these values.
00:01:15.894 --> 00:01:23.284
We can then see that the midpoints are going to be negative 3.5, π₯ equals negative 2.5, negative 1.5, and so on.
00:01:23.674 --> 00:01:25.764
Now, weβre going to need to be extra careful here.
00:01:25.994 --> 00:01:29.724
We see that some of our rectangles sit below the π₯-axis.
00:01:29.994 --> 00:01:32.804
This means weβre going to find the negative value of their area.
00:01:33.064 --> 00:01:41.644
In other words, weβll subtract the total area of the rectangles that sit below the π₯-axis from the total area of the rectangles that sit above the π₯-axis.
00:01:41.854 --> 00:01:44.954
Letβs begin by finding the function values at each midpoint.
00:01:44.954 --> 00:01:46.894
And that will tell us the height of each rectangle.
00:01:47.084 --> 00:01:52.844
Thatβs π of negative 3.5, π of negative 2.5, π of negative 1.5, and so on.
00:01:53.284 --> 00:01:55.944
And, of course, our function is π₯ squared minus four.
00:01:56.154 --> 00:02:08.924
And when we substitute each of these values into that function, we get 8.25, 2.25, negative 1.75, negative 3.75, another negative 3.75, and another negative 1.75.
00:02:09.524 --> 00:02:14.324
We then obtain the area of our first rectangle here to be 8.25 times one.
00:02:14.754 --> 00:02:18.084
Our second rectangle has an area of 2.25 times one.
00:02:18.564 --> 00:02:24.644
Our third rectangle has an area of 1.75 times one, not negative 1.75.
00:02:24.644 --> 00:02:26.614
Because weβre just dealing with areas at the moment.
00:02:26.854 --> 00:02:30.504
Our fourth rectangle has an area of 3.75 times one.
00:02:30.894 --> 00:02:35.774
Base times height with our fifth rectangle is 3.75 times one again.
00:02:35.984 --> 00:02:39.144
And for our last rectangle, itβs 1.75 times one.
00:02:39.774 --> 00:02:50.754
The Riemann sum is therefore 8.25 plus 2.25 minus the sum of 1.75, 3.75, 3.75, and another 1.75.
00:02:51.064 --> 00:02:58.194
And that gives us an approximation to the definite integral between the values of negative four and two of π₯ squared minus four.
00:02:58.354 --> 00:03:00.504
Itβs negative 0.5.
00:03:00.824 --> 00:03:04.574
And, of course, π₯ squared minus four is a fairly simple function to integrate.
00:03:04.824 --> 00:03:07.924
So we can check our answer by evaluating that integral.
00:03:08.294 --> 00:03:15.694
When we do, we get π₯ cubed over three minus four π₯ between the limits of negative four and two, which gives us a value of zero.
00:03:15.694 --> 00:03:18.614
And our estimate of negative 0.5 is pretty close.
00:03:18.794 --> 00:03:20.594
So we can assume weβve probably done this correctly.
00:03:21.324 --> 00:03:24.924
Itβs worth noting that a sketch of the curve wonβt always be possible.
00:03:25.174 --> 00:03:32.604
So instead, we need to notice that when the value of π of π₯ is less than zero, we subtract the area of the rectangle with that height.