Question

Prove that the volume of any paraboloid is always half the volume of the circumscribed cylinder?

Answer 1

Hint: To solve this problem we have to first calculate the volume enclosed by the paraboloid and then calculate the volume enclosed by the cylinder. Then compare the volumes of the solids to find the relation between paraboloid and cylinder.

Complete step by step solution:
As per the problem we have to prove that the volume of any paraboloid is always half the volume of the circumscribed cylinder.
First we have to calculate the volume enclosed by the paraboloid,
The volume enclosed by a surface of revolution of positive curve around the y axis is given as,
$V_1=\pi \underset{a}{\overset{b}{\int }}{\left(f\left(y\right)\right)}^{2}dy$
The lower limit is obvious as the extreme of the surface which is at $y=0$ this is when both x and y are zero. While the upper limit is unspecified as the problem asks to prove the formula for any height of paraboloid. The formula of parabola will be given as,
$y=c{x}^2$
On rearranging we will get,
$x=\sqrt{{\displaystyle \frac{y}{c}}}$
Now using the formula of the paraboloid to find the volume is equal to,
$V_1=\pi \underset{0}{\overset{h}{\int }}{\left(\sqrt{{\displaystyle \frac{y}{c}}}\right)}^{2}dy=\pi \underset{0}{\overset{h}{\int }}{\displaystyle \frac{y}{c}}dy$
Now on integrating we will get,
$V_1={\displaystyle \frac{\pi }{c}}{\left[{\displaystyle \frac{1}{2}}{y}^2\right]}_0^h={\displaystyle \frac{\pi h^2}{2c}}$
Now on calculating the volume of the cylinder,
The volume of a cylinder is the multiplication of height by the area of its circular cross-section. The radius is given by the rearranged parabola formula,
$x=\sqrt{{\displaystyle \frac{y}{c}}}$
Where,
$x=r$
$y=h$
$r=\sqrt{{\displaystyle \frac{h}{c}}}$
Thus the volume of the cylinder,
$V_2=h\pi {\left(\sqrt{{\displaystyle \frac{h}{c}}}\right)}^2=h\pi {\displaystyle \frac{h}{c}}={\displaystyle \frac{\pi h^2}{c}}$
Now equating both the volumes we will get,
$V_1=V_2$
${\displaystyle \frac{\pi h^2}{2c}}={\displaystyle \frac{\pi h^2}{c}}$
Here we can see that the volume of any paraboloid is always half the volume of the circumscribed cylinder.
Hence proved.

Note:
Remember that the paraboloid has equation which is equals to $y=c\left(x^2+z^2\right)$ where z is the axis that is coming out of the page and it is the surface that is revolving around the y-axis of the curve $y=c{x}^2$. Here we have used the technique of surface revolution for paraboloid.