More on Volumes of Revolution 4

Upon reflecting the curve $f(y) = \dfrac{25}{64}y^2$ on the interval $[\dfrac{-8}{5}\sqrt{\dfrac{\sqrt{5} - 1}{2}}, \dfrac{8}{5}\sqrt{\dfrac{\sqrt{5} - 1}{2}}]$ about the line $x = \dfrac{\sqrt{5} - 1}{2}$ we obtain the curve $g(y) = \sqrt{5} - 1 - \dfrac{25}{64}y^2$ .

Let $j = \sqrt{5} - 1$ and $a = \dfrac{8}{5}\sqrt{\dfrac{j}{2}}$ .

The volume $V = \pi \int_{-a}^{a} (g(y))^2 - (f(y))^2 dy = \pi \int_{-a}^{a} (j - \dfrac{25}{64}y^2)^2 - (\dfrac{25}{64})^2 y^4 dy =$ $\pi j \int_{-a}^{a} (j - \dfrac{25}{64}y^2) dy = \pi j (jy - \dfrac{25}{96}y^3)|_{-a}^{a} =$ $2\pi aj(j - \dfrac{25}{96}a^2) =$ $2\pi aj(j - \dfrac{2}{3}j) = \dfrac{4}{3}j^2a\pi =$

$\dfrac{16}{15}\sqrt{2} j^{\dfrac{5}{2}}\pi = \dfrac{16}{15}\sqrt{2}(\sqrt{5} - 1)^{\dfrac{5}{2}}\pi =$ $\dfrac{2^4}{5 * 3}\sqrt{2}(\sqrt{5} - 1)^{\dfrac{5}{2}}\pi = \dfrac{a^4\sqrt{a}}{bc}(\sqrt{b} - 1)^{\dfrac{b}{a}}\pi \implies a + b + c = \boxed{10}$ .