SOLUTION: suppose you estimate the area under a curve using inscribed rectangles and then estimate the area using circumscribed rectangles. how will the two estimates for the area compare?
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Question 426974: suppose you estimate the area under a curve using inscribed rectangles and then estimate the area using circumscribed rectangles. how will the two estimates for the area compare? Answer by robertb(5830) (Show Source):
You can put this solution on YOUR website! Let = sum of the areas of the inscribed rectangles (also called the lower Riemann sum), and
= sum of the areas of the circumscribed rectangles (also called the upper Riemann sum).
Then if the curve over the interval [a,b], (and using the same partitioning of the interval) then . As the partitioning gets finer (i.e., more subintervals), the two sums should converge to a common value which is the definite integral of f(x) over [a,b] if y = f(x) is integrable over [a,b]. (Note that if then the relationship between the two Riemann sums is also reversed.)
