Page:Calculus Made Easy.pdf/242

${\begin{aligned}{\text{Hence}}\;{\text{volume sphere}}&=2\int _{x=0}^{x=r}\pi (r^{2}-x^{2})\,dx\\&=2\pi \left[\int _{x=0}^{x=r}r^{2}\,dx-\int _{x=0}^{x=r}x^{2}\,dx\right]\\&=2\pi \left[r^{2}x-{\frac {x^{3}}{3}}\right]_{0}^{r}={\frac {4\pi }{3}}r^{3}.\end{aligned}}$

(2) Find the volume of the solid generated by the revolution of the curve $y^{2}=6x$ about the axis of $x$, between $x=0$ and $x=4$.

The volume of a strip of the solid is $\pi y^{2}dx$.

${\begin{aligned}{\text{Hence}}\;{\text{volume}}&=\int _{x=0}^{x=4}\pi y^{2}\,dx=6\pi \int _{x=0}^{x=4}x\,dx\\&=6\pi \left[{\frac {x^{2}}{2}}\right]_{0}^{4}=48\pi =150.8.\end{aligned}}$

 

On Quadratic Means.

In certain branches of physics, particularly in the study of alternating electric currents, it is necessary to be able to calculate the quadratic mean of a variable quantity. By “quadratic mean” is denoted the square root of the mean of the squares of all the values between the limits considered. Other names for the quadratic mean of any quantity are its “virtual” value, or its “R.M.S.” (meaning root-mean-square) value. The French term is valeur efficace. If $y$ is the function under consideration, and the quadratic mean is to be taken between the limits of $x=0$ and $x=l$; then the quadratic mean is expressed as

${\sqrt[{2}]{{\frac {1}{l}}\int _{0}^{l}y^{2}\,dx}}$.