#### To determine

**To find:**

i. x-coordinates of the points of intersection

ii. the volume of the solid

#### Answer

i. x=0 and x=2.175

ii. V≈14.45unit3

#### Explanation

**1) Concept:**

Let S be a solid.

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

V= ∫ab2πxf(x)dx

where 0≤a≤b, where f(x) is the height of the solid.

**2) Given:**

y=x2-2x, y=xx2+1

**3) Calculations:**

Given,

y=x2-2x, y=xx2+1

First, draw the graph of the given curves.

i. From the graph of the given curves,

The x coordinate of the points of intersection are x=0 and x≈2.175

ii. Also,x2-2x<xx2+1 on the interval (0, 2.175). Therefore, the height of typical shell is x2-2x-xx2+1

So, the volume of the solid rotating about the y-axis is

V= ∫02.1752πxxx2+1 -x2-2xdx

V= ∫02.1752πx2x2+1 -(x3-2x2) dx

V= 2π∫02.175x2x2+1 -(x3-2x2) dx

By using a calculator,

V≈14.45

Therefore, the volume of the solid about y-axis is ≈14.45 unit3

**Conclusion:**

The x- coordinate of the points of intersection are x=0 and x=2.175

The volume of the solid about y-axis is ≈14.45unit3