The volume generated by rotating one branch of hyperbola
Find the volume generated by rotating one branch of hyperbola
$x^2-y^2=a^2$
about the $x$-axis, between the limits $x=0$ and $x=2a$ using single-variable calculus/solid of revolution
My solution is: $V= \int \pi y^2~dx=\frac{2a^3\pi}{3}$ While in the solution manual it’s $\frac{4a^3\pi}{4}$
Solution 1:
The tricky point here is that there is nothing between $x=0$ and $x=a$. You can draw a graph of the hyperbola to discover this issue.
Just integrating between $x=a$ and $x=2a$ will give the right answer:
$$V=\int_a^{2a}\pi(x^2-a^2)\mathrm{d}x=\frac43\pi a^3$$