Three results for finding the area under the curve y = 9 - x^2, between x = 2 an

Question

Three results for finding the area under the curve y = 9 - x^2, between x = 2 and x = 3, are shown below. The result found by dividing the interval into 10 subintervals and then adding up the areas of the inscribed rectangles is 2.415. The result found by dividing the interval into 10 subintervals and then adding up the areas of the circumscribed rectangles is 2.915. The exact result found by evaluating the antiderivative at the bounds is approximately 2.67. Why is the mean of the sums of the inscribed rectangles and circumscribed rectangles less than the exact value?

Explanation / Answer

Inscribe rectangle are the rectangles which are completely inside the curve wheareas circumscribed rectangles are th erectangles which have area even outside the curve.
If we are using the inscribed rectangle method,we are missing all those rectangles which are not completedly inside but part of which contribute to the area under the curve.
Method of antiderivative does not miss out any portion while calculating the area because it divides the interval between infinitely small parts.So area is accurate.

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