(9) Find the volume generated by a sine curve revolving about the axis of ${\displaystyle x}$. Find also the area of its surface.

(10) Find the area of the portion of the curve ${\displaystyle xy=a}$ included between ${\displaystyle x=1}$ and ${\displaystyle x=a}$. Find the mean ordinate between these limits.

(11) Show that the quadratic mean of the function ${\displaystyle y=\sin x}$, between the limits of ${\displaystyle 0}$ and ${\displaystyle \pi }$ radians, is ${\displaystyle {\tfrac {\sqrt {2}}{2}}}$. Find also the arithmetical mean of the same function between the same limits; and show that the form-factor is ${\displaystyle =1.11}$.

(12) Find the arithmetical and quadratic means of the function ${\displaystyle x^{2}+3x+2}$, from ${\displaystyle x=0}$ to ${\displaystyle x=3}$.

(13) Find the quadratic mean and the arithmetical mean of the function ${\displaystyle y=A_{1}\sin x+A_{1}\sin 3x}$.

(14) A certain curve has the equation ${\displaystyle y=3.42\epsilon ^{0.21x}}$. Find the area included between the curve and the axis of ${\displaystyle x}$, from the ordinate at ${\displaystyle x=2}$ to the ordinate at ${\displaystyle x=8}$. Find also the height of the mean ordinate of the curve between these points.

(15) Show that the radius of a circle, the area of which is twice the area of a polar diagram, is equal to the quadratic mean of all the values of r for that polar diagram.

(16) Find the volume generated by the curve ${\displaystyle y=\pm {\dfrac {x}{6}}{\sqrt {x(10-x)}}}$ rotating about the axis of ${\displaystyle x}$.