By Wilson E.B.
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Extra resources for Advanced Calculus
Apply the considerations of Ex. 24 to rinding moments of inertia of a with respect to the origin, 0, x mx, y (a) a uniform triangle y = = a uniform rectangle with respect to the center, (7) a uniform ellipse with respect to the center. (|3) 26. Compare Exs. 24 and 16 to treat the volume under the surface z = p (x, y) and over the area bounded by y =/(x), y = 0, x = a, x = b. , y 0. x = 0, x = Q, (/3) under z (7) under \- a2 62 -\ = 1 c2 and over f r2 d< 27. Discuss sectorial area | of small sectors 28.
If there is an infinity of points in the set, possibilities arise. Either 1 far to the right that which case the set it is not possible to assign a point set is farther to the right one two K so no point of the said to be unlimited above is -or 2 in there a is K K such that no point of the set is beyond and the set is point said to be limited above. Similarly, a set may be limited below or unIf a set is limited above and below so that it is entirely contained in a finite interval, it is said merely to be limited.
Integrate 2 (/S) , of these curves = a, y (|8) by converting . ' 10. 2 : 2 BinSx ^^ Jf a 2cos2 x &2 sin z a; + ; L (z 1 2) Find the areas ' . ' 1 + 2 xm + tan- 2 xsecn x, ~ by trigonometric or hyperbolic substitution 2 b - 1) a n ' z (x 6. Integrate r J a x P 1 by decomposition /\ r x r 1 I = / 5. Integrate (a) a2 )" m tan'"- 1 x sec"x = = xm r . (5) tion 3 cosx]. lr. I 9. 4 sinx 2 cosx) -f Prove by integration by parts the reduction formulas r sin. m + l xcos- l x n smm x cos' x = sin m a; cos"- 2 x, (a) J m+n m + n J/ , 7.