5.1 Approximating Areas

[STARTS AT 17:05]

PROFESSOR: Find the lower sum of the function f of x equals sine x over the interval 0 to pi over 2. Based on what we know about sine from 0 to pi, it's greater than 0. And from pi to 2 pi, it's less than 0. 

So pi over 2 is this region. So we're finding that area. Because it's increasing, the left-hand sums are going to be the lower sum. So let's start with x0. That's our a value. That is 0. And our delta x, we need that to find the next function value. That would be pi over 2 minus 0 over 6, which is pi/12. 

So x1 is pi/12. x2 is 2 pi/12, so pi/6. That would be 3 pi/12, which would be pi/4. We have 6 integrals. So once we hit pi/2, we know we're there. That would be 4 pi/12, so pi/3, 5 pi/12, and 6 pi/12, or pi/2. So there is our last endpoint. 

Now, I am not going to actually evaluate these. We have sine of 0, sine of pi/12, f of x 2, sine of pi/6, sine of pi/4-- I'll put that value there. f of x5 is sine of 5 pi/12. And because we're doing a left-hand sum, we actually don't need that last function value. All right, but that is going to be 1 there. 

So our L6, our lower some with six rectangles-- i equals 1. No, starting at 0 going to 5 f of xi delta x. Our delta x is pi/12. It's the sum from 0 to 5 is sine of 0 plus sine of pi/12 plus sine of pi/6 plus sine of pi/4 plus sine of pi/3, sine of 5 pi/12. And that sum will be approximately 0.863. So there's my approximate area.

[ENDS AT 20:48]