Find sample points for endpoints

Question

Consider the following.

(a) Use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 24.

(i) Sample points are left endpoints.

L₆ = 

(ii) Sample points are right endpoints.

R₆ = 

(iii) Sample points are midpoints.

M₆ = 

Answer 1

Since the question wants us to use 6 rectangles and the entire range has a length of 24, each rectangle must have a width of 24 / 6, or 4. So we can just draw rectangles that are 4 units wide right up against each other. The vertices of these rectangles are at x = 0, 4, 8, 12, 16, 20, and 24. For the left endpoint, we can pick the left vertex of each rectangle (0, 4, 8, 12, 16, and 20). That makes are sample points f(0), f(4), f(8), f(12), f(16), and f(20).

Can you extend this logic to find the sample points for the right endpoints and the midpoints?