00:02
We want to evaluate the integral from 0 to 7 of the function f by interpreting it in terms of areas in the figure.
00:12
The areas of the labeled regions are a1 equal 8 here, blue on the left, a2 equal 3 right here, the yellow area on the left, and a3 equal 1 here, and a4 here equal 1 also.
00:34
So, knowing those level areas, we want to find the integral from 0 from here to here.
00:49
And we see that we get to separate that integral into three integrals, and each sub -integral gets to be considered in the regions labeled here a1, a2, a3.
01:03
That is, the integral from 0 to 7 of the given function f is equal to, the first integral goes from 0 to 3, and we get to put that separate, because on the interval 0, 3 the function is positive all the way through that interval, or 0 at the endpoints, which is not important for the value of the integral, plus the integral from 3 to 5, that is, on this interval, and we put that integral separated, because the function there is negative, plus the integral from 5 to 7 of f, that is, the integral over this interval.
01:58
And, again, we get to consider that integral separated, because the function there is always positive.
02:06
Good, so we know that the value, the actual value of the integral in each of these sub -intervals is just the area of the region.
02:18
But, because we are calculating the integrals without any actual value at all, we know that where the function is positive, the integral is a positive number, so in that case, the area is exactly equal to the integral, but where the function is negative, the integral is equal to the area with the negative sign...