Question

A solid lies between planes perpendicular to the​ y-axis at
`y=0` and
`y=4`. The​ cross-sections perpendicular to the​ y-axis are circular disks with diameters running from the​ y-axis to the parabola
`x=√(10) y^2`. Find the volume of the solid.

Answer 1

For any given
`0\le y\le4`, a cross section in that plane is a circle whose diameter is
`x=√(10)\,y^2` with thickness
`\Delta y`. Such a cross section contributes a volume of
`\pi \left(\frac x2\right)^2 \, \Delta y = \frac{5\pi}2 y^4 \, \Delta y`.

As
`\Delta y\to0` and the number of cross sectional cylindrical disks increases without bound, the infinite sum of their volumes converge to the volume of the solid, given by the definite integral

`\displaystyle \frac{5\pi}2 \int_0^4 y^4 \, dy = \frac{5\pi}2 \cdot \frac{4^5}5 = \boxed{512\pi}`