3 results for finding the area under the curve y=x^2 between x=0 and x=2 are sho

Question

3 results for finding the area under the curve y=x^2 between x=0 and x=2 are shown below. -the results found by divining the interval into 10 subintervals and then adding up the areas of the inscribed rectangles is 2.28 -the results found by divining the interval into 10 subintervals and then adding up the areas of the circumscribed rectangles is 3.08 -the exact result found by evaluating the antiderivative at the bounds is approximatly 2.67 why is the mean of the sums of the inscribe rectangles and circumscribed rectangles greater than the exact values?

Explanation / Answer

the integral and the area under a curve. Using the formulas for summing up the first n integers, the first n squares, the first n cubes, etc. we saw that we can compute the area bounded by any polynomial curve. For example, in class we used inscribed and circumscribed rectangles to compute the area bounded by y = 2x - x^2 on the interval [0,1]. Later we will learn that areas can be computed using antiderivatives. To find the area under y = 2x - x^2, we find an antiderivative for this function --- x^2 - x^3/3 in this case --- and then we evaluate the antiderivative at the endpoints:

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