Integration of $|f(x)|$ I would like to learn how to use the sign function $\operatorname{sgn}(x)$ to integrate the absolute value of an arbitrary function. I know that integrating piecewise should give me the same result, but sometimes trigonometric functions can become cumbersome. I have tried searching on the internet, but I can't seem to find anything. Perhaps the theory isn't complete, and the methods for integration aren't known for the absolute value of all functions, but I would appreciate if someone could give me one or two places where I can learn a general approach. What I would like to learn is integration for the absolute value of at least a second degree function, and the "basic" trigonometric functions (sin, cos, tan, csc, sec, cot). Links or references to information on the internet will be a lot more helpful to me than references to books. Thank you.
 A: I looked through some of my stuff last night and found the following references. However, all except one are not "links or references to information on the internet" (unless you have access to a university library), but perhaps by googling some of the titles and names you can come up with some useful items on the internet. I've listed them in the order (highest to lowest) that I'd guess as to how useful they'd be to you (with the exception of the last one, for which I've only seen the Nature summary).
Kenneth O. May, "The calculus of absolute values", American Mathematical Monthly 62 #9 (November 1955), 651-653. (here)
Margaret R. Wiscamb, "A formula for the derivative of the absolute value of a polynomial", Pi Mu Epsilon Journal 4 #9 (Fall 1968), 367-369.
C. Walmsley, "On the area of a square", Mathematical Gazette 38 #325 (September 1954), 210-211. (here)
John L. Spence, "Step function notation", School Science and Mathematics 60 #3 (March 1960), 179-180. (here)
Daryl E. Doan and Joshua Grusd, "Derivative and indefinite integral", Reader Reflections column, Mathematics Teacher 98 #5 (Dec./Jan. 2004/2005), 292.
"prove d|x|/dx = x/|x| without piecewise", MathKB discussion groups, Mathematics - General Topics, 27 January 2006. (here)
Nature (journal) 93 #2315 (12 March 1914), p. 39: Midway down the left column is a brief description of a Russian paper by I. N. Kouchnereff and D. Riabouchinsky that: "...treats this quantity [absolute value operation] as a function of the variable $x$, and shows how this method leads to interesting formulae involving the solution of equations, differentiation, and integration. A number of elegant geometrical applications are also given, such as equations of broken lines [polygonal curves?], and equations of limited portions [bounded?] of planes, such as a square area."
