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

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

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

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

(2) Find the area of the parabola ${\displaystyle y=2a{\sqrt {x}}}$ between ${\displaystyle x=0}$ and ${\displaystyle x=a}$. 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 ${\displaystyle y=\sin ^{2}x}$, and find the mean ordinate.

(5) Find the area included between the two branches of the curve ${\displaystyle y=x^{2}\pm x^{\tfrac {5}{2}}}$ from ${\displaystyle x=0}$ to ${\displaystyle x=1}$, 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 ${\displaystyle r}$, and of height ${\displaystyle h}$.

(7) Find the area of the curve ${\displaystyle y=x^{3}-\log _{\epsilon }x}$ between ${\displaystyle x=0}$ and ${\displaystyle x=1}$.

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