To determine

To set up:

An integral for the volume of a solid.

Answer

V=∫01π2-x22-2-x2dx

Explanation

1) Concept:

If the cross section is a washer with the inner radius rin and outer radius rout, then the area of the washer is obtained by subtracting the area of the inner disk from the area of the outer disk.

A=π outer radius2 -π inner radius2

Let  S  be a solid that lies between x=a and  x=b. If the cross sectional area of S in the plane Px through x and perpendicular to the x-axis, is  A(x), where A is continuous function, then the volume of S is

V=limn→∞⁡∑i=1nAxi*∆x=∫abAxdx 

2) Given:

y=x, y= x2;about y=2

3) Calculation:

The equation of given curve is

y=x, y= x2;about y=2 

Draw the curves and line  y=2 .

The curves y=x& y=x2 intersect at the points 0,0&1,1.

The cross section is perpendicular to y=2

The cross section in plane Px has the shape of a washer with the inner radius y=2-x  and outer radius y=2-x2

So, find the cross sectional area by subtracting area of the outer circle from the area of the inner circle, i.e.,

Ax=π(outer radius)2-π(inner radius)2

Ax=π2-x22-π2-x2

Ax=π2-x22-2-x2

Therefore,

V=∫01Axdx

V=∫01π2-x22-2-x2dx

Conclusion:

V=∫01π2-x22-2-x2dx