The plane figure bounded by the cardioid $r_1=2α(1+cos\ θ)$ and the parabola $r_2=\frac{2α}{1+cos\ θ}$, rotates around the polar axis. Show that the volume generated is $18πa^3$.
So the plane i have to consider is the following:
The volume generated is given by: $$ V=\int_{θ_1}^{θ_2}\frac{2}{3}πr^3\ sin\ θ\ dθ $$
Now, as far as i'm concerned i will need to break the integral into two parts. I can calculate the integral of $r_2$ from $θ=0$ to $θ=π/2$ and then the integral of $r_1$ from $θ=π/2$ to $θ=π$. So i will have to calculate this:
$$ V= \int_{0}^{π/2} \frac{2π}{3}\cdot \frac{8\ sin\ θ}{(1+cos\ θ)^3}\ dθ + \int_{π/2}^{π} \frac{2π}{3}\cdot 8(1+cos\ θ)^3 sin\ θ \ dθ $$
And in doing so i get $10πα^3/3=10.472α^3$. And not $18πα^3$ as i should.
Where am i wrong? Any help will be greatly appreciated. Thanks in advance!