00:02
Okay, 5 .22, here is our introduction to definite integrals, and we're using the limit definition of an integral.
00:11
So this gets a little bit quite complicated with the algebra.
00:14
It gets easier once we leave this behind and use integration rules.
00:17
But to understand the theory of what's going on and test your ability to stick with a problem, we're using this algebraic formulation.
00:24
So you were given a theorem four in this section that said, hey, the definite integral is expressed by the limit as n goes to infinity of.
00:31
This is just the sum of the area of rectangles.
00:35
So we're going to have n as a number of rectangles, let n go to infinity.
00:39
And so delta x, the width of each rectangle, is just going to be the difference between the limits of integration divided by n.
00:47
And then x sub i, these are the heights of the rectangle.
00:51
So we're going to step along and say, ok, starting at the left boundary of integration, and then increment one delta x, 2 delta x, 3 delta x, until we get there.
01:00
Now, in this problem, i find it helpful just to break this up into three different integrals.
01:05
It breaks the algebra up into different pieces.
01:08
So if we can write this as the integral from 1 to 4 of x squared dx minus the integral of 1 to 4, 4x dx, plus the integral 1 to 4 of 2 dx.
01:24
And now we're going to come back there later.
01:26
So we're going to work on each individual piece, and then we're going to come right back to this spot.
01:32
For our final answer.
01:38
So let's work on the first piece.
01:40
So the integral from 1 to 4 of x squared d x.
01:49
So this is going to be the limit as n approaches infinity of the sum of i equals 1 to n.
02:02
And now you look at the function is x squared.
02:06
So we're going to have the limit of integration starts at 1.
02:09
So you're going to have one plus.
02:12
Now, if you look at what is delta x? so in this case, delta x is 4 minus 1 over n, which is 3 over n.
02:22
So this is 1 plus 3i over n squared, and then the width of each rectangle is simply 3 over n.
02:35
So this is the limit as n approaches infinity.
02:43
And let's just bring this over here of 3 over n, the sum i equals 1 to n.
02:53
And now let's just square this binomial.
02:55
So this is going to be 1 plus 6i over n plus 9 i squared over n squared.
03:19
This is the limit as n approaches infinity of 3 over n.
03:28
And now let's just take a look and see what we have here.
03:32
The sum from i equal 1 to n of 1 is just simply going to be n.
03:37
So this is going to be n plus 6.
03:46
The sum from i equal 1 to n, so the formula is the sum from i equal 1 to n of i.
03:54
It's just simply n, n plus 1 over 2.
03:58
So this becomes n, n plus 1 over 2n, plus, and then the sum, i equal 1 to n, of i squared, is equal to n, n plus 1, 2n plus 1 over 6...