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Absolutely Different (Posted on 2008-07-24) Difficulty: 2 of 5
Evaluate this definite integral:

pi
∫ (| |sin y| - |cos y| |) dy
0

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

  Submitted by K Sengupta    
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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)

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(3): Some Thoughtsbrianjn2008-07-25 00:18:35
re(2): Some ThoughtsBractals2008-07-24 18:33:32
Solutiondivide et imperaAdy TZIDON2008-07-24 13:18:50
re: Some ThoughtsSteve Herman2008-07-24 12:50:58
Some ThoughtsSome ThoughtsBractals2008-07-24 12:14:16
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