#### To determine

**To estimate:**

The volume obtained by rotating about the y-axis the region under the given curve.

#### Answer

V=3.6807

#### Explanation

**1) Concept:**

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

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= ∫ab2πxf(x)dx

where, 0≤a<b

iii. To approximate the integral is the 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. And xi=a+i∆x

**3) Given:**

y=1+x3, 0≤x≤1

**4) Calculation:**

As the region is bounded by

y=1+x3, 0≤x≤1

Draw the region using the given curves.

The graph shows the region and the height of cylindrical shell formed at x by the rotation about the line y-axis

It has the radius x.

Therefore, the circumference is 2πx and the height is 1+x3

It is given that 0≤x≤1

So, a=0 and b=1

So, the volume of given solid is

V=∫012πxfxdx

=∫012πx1+x3dx

Assume that

V=∫01gxdx

where gx=2πx1+x3

For the midpoint rule,

Find ∆x &xi-.

0≤x≤1, means a=0, b=1 and n=5

∆x=b-an=1-05=15=0.2

So, the subintervals are 0, 0.2, 0.2, 0.4, 0.4, 0.6, 0.6, 0.8, 0.8, 1

Then the midpoints will be 0.1, 0.3, 0.5, 0.7, 0.9

By using the midpoint rule, the volume is

∫01gxdx=∑i=15g(xi-)∆x

=g0.1+g0.3+g0.5+g0.7+g0.90.2

=0.6286+1.9102+3.3321+5.0970+7.43560.2

=18.40370.2

=3.6807

**Conclusion:**

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

∫012πx1+x3dx~3.6807