Estimate the area of the graph f(x) = √x from x=0 to x = 4 using four approximating rectangles and right endpoints. Is it an underestimate or overestimate?

Question

What is the answer using left endpoints?

Answer 1

y = sqrx

integral of y = (2/3)x^(3/2)

evaluated at x =4 is (2/3)(4)^(3/2) = (2/3)(4)2)= 16/3 = 5 1/3 = exact area under the curve from x=0 to x=4

4 rectangles would each have base = 1

endpoints would be 1, sqr2, sqr3 and 2

sum them: about 3+1.414 + 1.732 = 6.146 which > 5.333, an overestimate

4 rectangles with the left end points of 0, 1, sqr2 and sqr3

times base of 1, sums to 1+1.414 +1.732 = 3.333, an underestimate

4 rectangles with the midpoints: sqr(1/2), sqr(3/2), sqr(5/2) and sqr(7/2)

sum to .707 +1.225 + 1.581 + 1.871 = 5.384, which is very close to the exact area, but a slight overestimate.