Page:Calculus Made Easy.pdf/244

or $\quad \left[{\frac {a^{x}}{\log _{\epsilon }a}}\right]_{x=0}^{x=l}$, 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 $y=\sin ^{2}x$, 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 $x=4$.