Page:Calculus Made Easy.pdf/245

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

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

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

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

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

(14) A certain curve has the equation $y=3.42\epsilon ^{0.21x}$. Find the area included between the curve and the axis of $x$, from the ordinate at $x=2$ to the ordinate at $x=8$. 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 $y=\pm {\dfrac {x}{6}}{\sqrt {x(10-x)}}$ rotating about the axis of $x$.