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

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 rectangle is 12.15. he result found by dividing the interval into 10 subintervals and then adding up the areas of the circumscribed rectangle is 14.85. The exact result found by evaluating the ant derivative at the bounds is 13.5. Why is the mean of the sums of the inscribed rectangle and circumscribed rectangles equal to the exact value?

Explanation / Answer

Consider the inscribed rectangle , here we are neglecting some part of the area under the curve . The neglected part here is a right triangle.

Consider the circumscribed rectangle , here we are adding some extra part to the area under the curve . The added part here is a right triangle.

These two right triangles are actually the triangles formed when we draw a diagonal on the rectangle. Hence these areas must be equal.

so, mean of the sums of the inscribed rectangles and circumscribed rectangles equal to the exact value

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