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,

you are playing an adventure game on your computer. you come across a troll who won't let you pass and there is no other way to get to the other side of the river except over the bridge he is guarding.

the troll is a nerd. he takes great delight in watching people struggle with his problems. very few get them. he only gives you one chance and then he changes the puzzle. you have to solve this one.

before you are four blocks. they are numbered 11,6, -7,21.

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