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)

step-by-step explanation:

first, eliminate the parenthesis:

3

x

+

2

−

4

x

second, group like terms:

3

x

−

4

x

+

2

third, combine like terms:

(

3

−

4

)

x

+

2

→

−

1

x

+

2

→

−

x

+

2

c

step-by-step explanation: