To determine

To express:

a in terms of  V

Answer

a=V2π+23

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 the line x=a from b to c is

V= ∫bc2π a-x f(x)dx

where  0≤b≤c

2) Given:

The triangle with the vertices (0, 0), (1, 0), (1, 2) rotated about the line x=a

3) Calculation:

The triangle with the vertices (0, 0), (1, 0), (1, 2) is shown in figure.

First, join the given points.

From the figure, the boundary curves are y=2x, y=0 and x=2

Therefore, the region of the triangle T is bounded by  y=2x and x=2 rotated about the line x=a.

Using the shell method, find the typical approximating shell is with radius (a-x),

Therefore, the circumference is 2π(a-x) and the height is y=2x

So, the total volume is

V= ∫ab2π(a-x) (2x)dx

V= 4π∫01ax-x2dx

V=4πax22-x3301

=4π a(1)22-133-0

V=4πa2-13

Solve the equation for a in terms of V.

V4π=a2-13

V4π+13=a2

a=2V4π+13

Therefore,

a=V2π+23

Conclusion:

 a in terms of V  is a=V2π+23