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 Absolutely Different (Posted on 2008-07-24)
Evaluate this definite integral:

pi
∫ (| |sin y| - |cos y| |) dy
0

Note: |x| denotes the absolute value of x.

 Submitted by K Sengupta No Rating Solution: (Hide) 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

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