Use the midpoint rule with n = 5 to estimate the volume v obtained by rotating about the y-axis the region under the curve y = 1 + 9x3 , 0 ≤ x ≤ 1. (round your answer to two decimal places.)

Using the shell method, the volume is given exactly by the definite integral,

Splitting up the interval [0, 1] into 5 subintervals gives the partition,

[0, 1/5], [1/5, 2/5], [2/5, 3/5], [3/5, 4/5], [4/5, 5]

with left and right endpoints, respectively, for the -th subinterval

where . The midpoint of each subinterval is

Then the Riemann sum approximating the integral above is

(compare to the actual value of the integral of about 14.45)

56.6

step-by-step explanation:

the ratio of the perimeters is equal to the scale.

the ratio of the areas is equal to the square of the scale.

34 / 20 = k

a / 19.6 = k²

substituting:

a / 19.6 = (34 / 20)²

a = 19.6 (34 / 20)²

a ≈ 56.6