The value that would be obtained using the 5 midpoint rectangles of equal width to estimate the given Integral is; 33/100
How to partition integral?
First of all, let us partition the interval [0, 1] as;
[0, 1/5] ∪ [1/5, 2/5] ∪ [2/5, 3/5] ∪ [3/5, 4/5] ∪ [4/5, 5]
Thus, the midpoints of the intervals which are the sample points are, respectively;
1/10, 3/10, 1/2, 7/10, 9/10
The integral is approximated by;
(0,1)∫x² dx = (5, n=1)Σf(x_n) Δx
where Δx is the difference between the partition endpoints, i.e.
Δx = (1 - 0)/5 = 1/5
and x_n is the midpoint of the nth partition.
Thus, we have;
(5, n=1)Σf(x_n) Δx = (1/5)[(1/10)² + (3/10)² + (1/2)² + (7/10)² + (9/10)²] = 33/100
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