Absolutely Different (Posted on 2008-07-24)

	Submitted by K Sengupta
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Let, f(y) = ||sin y|- |cosy|| Then, we observe that: f(pi-y) = f(pi/2 - y) = f(y) Hence, we must have: Integr.{0 to pi} f(y) dy = 2*Integr.{0 to pi/2} f(y) dy and, Integr.{0 to pi/2} f(y) dy = 2*Integr.{0 to pi/4} f(y) dy Accordingly, we have: Integr.{0 to pi} f(y) dy = 4*Integr.{0 to pi/4} f(y) dy Now, we know that: tan y < 1, whenever 0 < y < pi/4 or, cosy > sin y, whenever y < pi/4 Hence, we must have: Integr.{0 to pi/4} f(y) dy = Integr.{0 to pi/4} (cosy - siny) = (sin y + cos y){0 to pi/4} = 1/√2 + 1/√2 - 1 = √2 - 1 Consequently, the value of the required definite integral is 4(√2 - 1)

	Subject	Author	Date
	re(3): Some Thoughts	brianjn	2008-07-25 00:18:35
	re(2): Some Thoughts	Bractals	2008-07-24 18:33:32
	divide et impera	Ady TZIDON	2008-07-24 13:18:50
	re: Some Thoughts	Steve Herman	2008-07-24 12:50:58
	Some Thoughts	Bractals	2008-07-24 12:14:16