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 $$\tfrac{\sqrt2}{2}$$. 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 $$x$$.