Page:Calculus Made Easy.pdf/244

 or $$\quad \left[ \frac{a^x}{\log_\epsilon a} \right]^{x=l}_{x=0}$$, which is $$\dfrac{a^l-1}{\log_\epsilon a}$$.

Hence the quadratic mean is $$\sqrt[2] {\dfrac{a^l - 1}{l \log_\epsilon a}}$$.

Exercises XVIII. (See p. 263 for Answers.)

(1) Find the area of the curve $$y=x^2+x-5$$ between $$x=0$$ and $$x=6$$, and the mean ordinates between these limits.

(2) Find the area of the parabola $$y=2a\sqrt x$$ between $$x=0$$ and $$x=a$$. Show that it is two-thirds of the rectangle of the limiting ordinate and of its abscissa.

(3) Find the area of the positive portion of a sine curve and the mean ordinate.

(4) Find the area of the positive portion of the curve $$y=\sin^2x$$, and find the mean ordinate.

(5) Find the area included between the two branches of the curve $$y=x^2\pm x^{\tfrac{5}{2}}$$ from $$x=0$$ to $$x=1$$, also the area of the positive portion of the lower branch of the curve (see Fig 30, p. 108).

(6) Find the volume of a cone of radius of base $$r$$, and of height $$h$$.

(7) Find the area of the curve $$y=x^3-\log_\epsilon x$$ between $$x=0$$ and $$x=1$$.

(8) Find the volume generated by the curve $$y=\sqrt{1+x^2}$$, as it revolves about the axis of $$x$$, between $$x=0$$ and $$x=4$$.