Question

Find the volume of the solid generated by revolving the region bounded by the lines and curves about the x-axis. y = x, y =1, x = 0

Answer 1

The volume is:
`V = (\pi)/(3)` cubic units

Explanation:

Volume of a solid:

The volume of a solid, given by the function f(x), over an interval between a and b, is given by:

`V = \pi \int_(a)^(b) (f(x))^2 dx`

y = x, y =1, x = 0

This means that the upper function is y = 1, and the lower function is y = x. So

`f(x) = (1 - x)`

The lower limit of integration is x = 0.

The upper limit is y = x when y = 1, so x = 1.

Then

`V = \pi \int_(a)^(b) (f(x))^2 dx`

`V = \pi \int_(0)^(1) (1-x)^2 dx`

`V = \pi \int_(0)^(1) (1-2x+x^2) dx`

`V = \pi (x-x^2+(x^3)/(3))|_(0)^(1) dx`

`V = \pi(1 - (1^2) + (1^3)/(3))`

`V = (\pi)/(3)`