Three results for finding the area under the curve y= 9_xf, between x=2 and x=3,

Question

Three results for finding the area under the curve y= 9_xf, 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

Solution:- Note that this is slightly less than the exact value we put above - this is because we 've used inscribed rectangles, and there are little gaps above each one not included in my sum. You could get this closer to the actual value by making each rectangle slightly thinner.

When we increase the number of rectangles (of equal width) used, using a smaller value for x ( = the width of the rectangles), we get a better approximation to the area.

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