Three results for finding the area under the curve y=3x, between x=0 and x=3, ar

Question

Three results for finding the area under the curve y=3x, between x=0 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 12.15.

The result found by dividing the interval into 10 subintervals and then adding up the areas of the circumscribed rectangles is 14.85

The exact result found by evaluating the antiderivative at the bounds is 13.5.

Why is the mean of the sums of the inscribed rectangles and circumscribed rectangles equal to the exact value?

Explanation / Answer

since this is a straight line and you are dividing the area into 10 equal rectangular parts, the small triangular area left out by each inscribed rectangle will be equal to the additional area covered by each circumscribed rectangle.

But, by integration, that area is very negligible, so you get almost the exact answer, which is the mean of the sums of the areas of the inscribed and circumscribed rectangles.

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