5.1 Approximating Areas

[STARTS AT 13:45]

PROFESSOR: Find a lower sum for 10 minus x squared on 1 to 2 with four subintervals. I'm going to sketch what that function looks like, so we can have an idea over the interval 1 to 2. It has a vertex of 0, 10. And it is decreasing to the right and increasing on the left. 

So from 1 to 2 is the area we're concerned with. If you notice, this is a decreasing function. So the lower sum is actually going to be coming from the right. So we have a lower function, or the lower sum is Rn, if it's decreasing. 

All right, so let's go ahead and work this out. So a, being x0, is 1. We need to know delta x. b minus a over n is 4, which then is going to be 1/4. So x1 is going to be 5/4. x2 is 6/4 or 3/2; 7/4 for x3; and 8/4 or 2, and that is our B value. 

Now, our function values-- 10 minus x squared, the first is 9. f of x1 is 8.4375. f of x2 is 7.75. f of x3 is 6.9375. And f of x4 is 6. So again, we're evaluating the function 10 minus x squared at each of these endpoints to get these function values. And what we're going to some is our function values. 

So in this problem, because we want to lower sum, what we are after is R4. That'd be the sum from 1 to 4 f of x i delta x, which will be 1/4 i equals 1 to 4 of our function values f of xi. So starting at 1 and going to 4, 8.4375 plus 7.75, 7.9375 plus 6. Finding 1/4 of that, that is 7.28125. And that is our lower sum.

[ENDS AT 17:00]