A solid lies between planes perpendicular to the y-axis at yequals0 and yequals2. The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola x equals StartRoot 6 EndRoot y squared. Find the volume of the solid.
Accepted Solution
A:
Answer:The volume of the solid is [tex]\frac{48\pi}{5}[/tex]Step-by-step explanation:Consider the provided information.The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola [tex]x=\sqrt6y^2[/tex]Therefore, diameter is [tex]d=\sqrt6y^2[/tex]Radius will be [tex]r=\frac{\sqrt6y^2}{2}[/tex]We can calculate the area of circular disk as: πr²Substitute the respective values we get:[tex]A=\pi(\frac{\sqrt6y^2}{2})^2[/tex][tex]A=\pi(\frac{6y^4}{4})=\frac{3\pi y^4}{2}[/tex]Thus the volume of the solid is:[tex]V=\int\limits^2_0 {\frac{3\pi y^4}{2}} \, dy[/tex][tex]V=[{\frac{3\pi y^5}{2\times 5}}]^2_0[/tex][tex]V=\frac{48\pi}{5}[/tex]Hence, the volume of the solid is [tex]\frac{48\pi}{5}[/tex]