#### To determine

**To estimate:**

The volume of solid using the midpoint rule.

#### Answer

V≈766.5 units3

#### Explanation

**1) Concept:**

i. If x is the radius of the typical shell, then the circumference =2πx and the height is y=f(x)

ii. By the shell method, the volume of the solid by rotating the region under the curve y=f(x) about x-axis from a to b is

V= ∫ab2πx f(x)dx

where 0≤a<b

iii. To approximate the integral is same as finding the Riemann sum for the midpoints. Use the formula and find the value of the integral.

**2) Formula:**

The Riemann sum for the midpoints:

Mn=∑i=1ng(xi-)∆x~∫abg(x)dx

where, xi-=12xi-1+xi and ∆x=b-an, n is the number of the subintervals.

**3) Given:**

n=5

**4) Calculation:**

The volume of the solid is

V= ∫ab2πx f(x)dx

Assume that, gx=2πx fx

So, volume of solid is

V= ∫abg(x)dx

where x is the radius and fx is the height of each cylinder.

So, fx=yi-yi-1

where yi is the upper part of the region and yi-1 is the lower part of the region

From the given figure,

The region is bounded between 0 to 10

That means a=0, b=10 and n=5

So the subintervals are 0, 2, 2, 4, 4, 6, 6, 8, 8, 10

Then the midpoints will be 1, 3, 5, 7, 9

∆x=b-an=10-05=2

By using the midpoint rule, the volume of the solid is

V=∑i=15g(xi-)∆x

=g1+g3+g5+g7+g92

Substitute gx=2πx fx,

=2π1f1+3f3+5f5+7f7+9f92

To write the values of fxi from the given figure,

=4π14-2+35-1+54-1+74-2+94-2

=4π2+12+15+14+18

=4π61

=244π

≈766.5 units3

**Conclusion:**

The volume of the solid obtained by rotating the given region about the y-axis is

V≈766.5 units3