More on Volumes of Revolution 3

$f(x) = g(x) \implies x^2 + x - 1 = 0 \implies x = \dfrac{\sqrt{5} - 1}{2} \implies$ the two curves intersect at $x = \dfrac{\sqrt{5} - 1}{2} \implies V_{1} = 2\pi \int_{0}^{\frac{\sqrt{5} - 1}{2}} x(\dfrac{4}{5}(\sqrt{x} + \sqrt{1 - x^2}) - \dfrac{8}{5}\sqrt{x}) dx =$ $\dfrac{8\pi}{5} \int_{0}^{\frac{\sqrt{5} - 1}{2}} x\sqrt{1 - x^2} - x^{\frac{3}{2}} dx =$ $\dfrac{8\pi}{5} (\dfrac{-1}{3} (1 - x^2)^{\frac{3}{2}} - \dfrac{2}{5} x^{\frac{5}{2}})|_{0}^{\frac{\sqrt{5} - 1}{2}} =$

$\dfrac{8\pi}{5} (\dfrac{1}{3} - (\dfrac{\sqrt{5} - 1}{2})^{\dfrac{3}{2}} (\dfrac{2 + 3\sqrt{5}}{15}))$ .

$h(x) = j(x) \implies x^2 + x - 1 = 0 \implies x = \dfrac{\sqrt{5} - 1}{2} \implies$ the two curves intersect at $x = \dfrac{\sqrt{5} - 1}{2} \implies V_{2} = 2\pi \int_{0}^{\frac{\sqrt{5} - 1}{2}} x(\dfrac{-8}{5}\sqrt{x} - \dfrac{4}{5}(\sqrt{x} - 3\sqrt{1 - x^2}))dx =$ $\dfrac{24\pi}{5} \int_{0}^{\frac{\sqrt{5} - 1}{2}} x\sqrt{1 - x^2} - x^{\frac{3}{2}} dx =$ $\dfrac{24\pi}{5} (\dfrac{-1}{3} (1 - x^2)^{\frac{3}{2}} - \dfrac{2}{5} x^{\frac{5}{2}})|_{0}^{\frac{\sqrt{5} - 1}{2}} =$

$\dfrac{24\pi}{5} (\dfrac{1}{3} - (\dfrac{\sqrt{5} - 1}{2})^{\dfrac{3}{2}} (\dfrac{2 + 3\sqrt{5}}{15})) = 3V_{1} \implies \dfrac{V_{2}}{V_{1}} = \boxed{3}$ .