To determine

To express:

V2 in terms of V1, k, and A

Answer

V2=V1+2πkA

Explanation

1) Concept:

i) Solid of Revolution:

Volume of solid after revolving a region around a line is given by

V=∫abA(x)dx or V=∫cdA(y)dy

In case of the disk method, radius is found in terms of x or y and use A=π·radius2

In case of the washer method, find the inner and outer radius and compute the area of washer by subtracting the area of inner disk from outer disk, use

A=π·outer radius2-π·inner radius2

ii) Area between curves:

The area A bounded by the curves y=f(x), y=g(x) and the lines x=a, x=b, where f and g are continuous and fx≥g(x) for all x in [a, b], is

A=∫ab[fx-g(x)]dx

2) Calculation:

Case 1:

According to the given condition, R is rotated about x axis to form a solid with volume V1 which forms the disk.

Now,

The area A bounded by the curves y=f(x), y=g(x) and the lines x=a, x=b, where f and g are continuous and fx≥g(x) for all x in [a, b], is

A=∫ab[fx-g(x)]dx    ….(1)

If cross section is solid, find radius of the solid in terms of y or x and use

A=π·outer radius2-π·inner radius2

A(x)=πf(x)2-g(x)2

Therefore,

V1=∫abA(x)dx=∫abπf(x)2-g(x)2dx     …..(2)

Case 2:

According to  the given condition, R is rotated about the line  y=-k  to form a solid with volume V2 which forms the washer.

So,

V2=∫abA(x)dx

If cross section is a washer, find the inner and outer radius from the sketch and compute the area of washer by subtracting the area of the inner disk from outer disk.

A=π· outer radius2-π· inner radius2

Ay=πfx+k2-gx+k2

V2=∫abA(x)dx=∫abπfx+k2-gx+k2dx

V2=∫abπfx2+k2+2kfx-gx2+k2+2kgx dx

V2=∫abπf(x)2-g(x)2+2πk∫ab[fx-g(x)]dx

From (1) and (2),

Therefore,

V2=V1+2πkA

Conclusion:

Therefore,

V2=V1+2πkA