Prove that if is positive and monotonic, then M N lies between R N and L N and is closer to the actual area under the graph than both R N and L N In the case that is increasing, Figure 18 shows that the part of the error in RN due to the ith rectangle is the sum of the areas A B D, and for MN it is...

and ba Suppose fx is monotonic increasing on the interval a b Ax N Jk0 x0 a a Ax a 2Ax ...

Prove that if is positive and monotonic, then M N lies between R N and L

Question:

Prove that if ƒ is positive and monotonic, then M_N lies between R_N and L_N and is closer to the actual area under the graph than both R_N and L_N. In the case that ƒ is increasing, Figure 18 shows that the part of the error in RN due to the ith rectangle is the sum of the areas A + B + D, and for MN it is |B − E|. On the other hand, A ≥ E.

1 A B C D E F X-1 Midpoint x; -x