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

Answer:

The volume is: $V = \frac{\pi}{3}$ cubic units

Step-by-step 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

$V = \pi \int_{a}^{b} (f(x))^2 dx$

$V = \pi \int_{0}^{1} (1-x)^2 dx$

$V = \pi \int_{0}^{1} (1-2x+x^2) dx$

$V = \pi (x-x^2+\frac{x^3}{3})|_{0}^{1} dx$

$V = \pi(1 - (1^2) + \frac{1^3}{3})$

$V = \frac{\pi}{3}$