Volumes of Solids of Revolution: Disc and Washer Methods

Find the volume when the line y = 3 from x = 0 to x = 5 is revolved around the x-axis.

V = pi * integral from 0 to 5 of (3)^2 dx = pi * [9x] from 0 to 5 = 45*pi cubic units.

Find the volume when y = x from x = 0 to x = 3 is revolved around the x-axis (verifying the volume of a cone).

V = pi * integral from 0 to 3 of x^2 dx = pi * [x^3 / 3] from 0 to 3 = 9*pi cubic units.

Find the volume when y = sqrt(x) from x = 1 to x = 9 is revolved around the x-axis.

V = pi * integral from 1 to 9 of x dx = pi * [x^2 / 2] from 1 to 9 = pi * (81/2 - 1/2) = 40*pi cubic units.

Find the volume when y = x^2 from x = 0 to x = 2 is revolved around the x-axis.

V = pi * integral from 0 to 2 of (x^2)^2 dx = pi * integral from 0 to 2 of x^4 dx = pi * [x^5 / 5] from 0 to 2 = (32/5)*pi cubic units.

Find the volume of the sphere formed by revolving the upper semicircle y = sqrt(r^2 - x^2) from x = -r to x = r around the x-axis.

V = pi * integral from -r to r of (r^2 - x^2) dx = pi * [r^2*x - x^3 / 3] from -r to r = (4/3)*pi*r^3 cubic units.

Find the volume generated by revolving the region enclosed by y = x and y = x^2 around the x-axis.

Intersection points are x = 0 and x = 1. Outer curve is y = x, inner curve is y = x^2. V = pi * integral from 0 to 1 of (x^2 - x^4) dx = pi * [x^3 / 3 - x^5 / 5] from 0 to 1 = pi * (1/3 - 1/5) = (2/15)*pi cubic units.