I know that to find the volume I am essentially finding the sum of the volume of small slices of the shape through integration. But I am having trouble applying this theory. Can somebody help run through the solution for this problem so I can look thoroughly at the steps.
The triangle with vertices (a,a),(a,2a),(2a,2a) is rotated
(a) about the x-axis
(b)about the y-axis.
Find the volume generated in each case.
For example, with the washer method, integrate on slices of rings with area $\pi(r_2^2-r_1^2)$.
a) the volume integral is
$$V= \int_a^{2a} \pi[(2a)^2- x^2]dx =\pi \int_a^{2a}(4a^2-x^2)dx $$ $$= \pi\left(4a^3-\frac13(8-1)a^3\right)=\frac{5\pi}{3}a^3 $$
b) the integral is
$$V= \int_a^{2a} \pi[y^2-a^2]dy$$ $$=\pi\left( \frac13(8-1)a^3-a^3\right) = \frac{4\pi}{3}a^3$$