00:01
How do you find the area under a particular function? what you do is you could break up the region into very narrow strips and add up the rectangular area, but to get more accurate, you want to do the integration, which is the area you would get if each of those strips had infinitesimal width, and there were an infinite number of them.
00:30
So let's take as an example, the area bounded by the following.
00:44
And we'll draw a sketch of that region.
00:50
So first of all, we have the curve, y of x equals 4, x squared plus 5.
00:59
But it is also bounded by y equals 0 and x equals 2.
01:10
Y equals 0 is the axis, the x -axis in particular.
01:20
X equals 2 is a line that we draw through x equals to 2, vertical line.
01:32
And then the curve, y of x, is a parabola that starts at 5 and just goes upwards.
01:47
So the area that we're after is we'll just draw it in there.
01:56
And like i said, you could break it up into strips, little rectangles, each of a finite width, and add up the length times width of each of those.
02:07
But a better way to do that is to take the y of x is the height of the rectangle.
02:18
Dx is the width, and we're going to let that collapse to zero, with the integral, and we are going to start with x equals 0 and go to x equals 2.
02:39
And now it's a matter of doing calculus that we can kind of differentiate backwards.
02:48
We have a polynomial and a reminder that if you had a polynomial such as a x to the n, and you took the derivative with respect to x, what you would do is drop down the x -best, in front as a multiplier and then drop the exponent by a factor of one.
03:12
And so doing the averse is what you do with the integral...