Problems in Differential Calculus Supplementary to a Treatise On Differential C

W E William Elwood Byerly

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V. Example 6.
20. Tcos^a-.
Ans.
1 sin^x — f sin^ic + sin a;.
21. / sin^j? cos^a?.
^715. 1 sin^a; — ^ sin^^.
29 r^in^.
JxCOS^X ^7is. Sec X -\- cos X.
23. Y>..
^7^^. (a; — l)e^.
Suggestion. Integrate by parts.
24. Tice"^.
^ . 471.. F^~i\- 25. / a; cos a?.
Ans. A; sin a; + cos x.
Definite Integrals.
26. Show that the area bounded by the curve g=f(x'), the axis of X, and the ordinates corresponding to x = Xq and x = Xi is Ly ^ Ax=xi IjJ ^ Ax=xq 27. If the curve y=f(x) lies on the positive side of t...he axis of X and y has neither a maximum nor a minimum value between the points (xq, t/o) and (xi, yi) show that if a set of n — 1 equidistant ordinates are erected between ?/o and y^ and with these as altitudes a set of inscribed rectangles and a set of circumscribed rectangles are constructed, the areas of the two sets will differ by the difference between the last rec- tangle of one set and the first rectangle of the other set ; that is, by the absolute value of (?/i — yo) Ax if Ax is — • Hence, show that the area of either set will approach the INTEGRATION.

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