Estimating the area under the curve using wapoint sum
when rounded to the nearest hundredth, the area under the graph of f(x) = x2 + 1 from x = 1 to x = 4 with 6
subintervals using the midpoint sum is
20.38
23.94
27.88
33.47

Splitting up the interval [1, 4] into 6 equally-spaced subintervals gives the partition

[1, 3/2], [3/2, 2], [2, 5/2], [5/2, 3], [3, 7/2], and [7/2, 4]

each with length (4-1)/6 = 1/2.

The endpoints of each subinteval form arithmetic sequences,

( for left, for right) with .

The midpoints of each subinterval are, respectively,

5/4, 7/4, 9/4, 11/4, 13/4, and 15/4

which are prescribed by the sequence,

( for midpoint) where .

The integral of over the interval [1, 4] is approximated by the sum of the areas of rectangles with length equal to the base of each subinterval and height equal to , the value of at the midpoint of that subinterval:

Compare to the actual value of the integral,

just keep asking questions until you have a better understanding.

step-by-step explanation: