Calculus problem #2195

Calculus Level 2

What is the value of ∫ 0 π ∣ sin ⁡ 2 x ∣ d x \displaystyle \int_{0}^{\pi} |\sin 2x|\ dx ?

Details and assumptions

The notation ∣ ⋅ ∣ | \cdot | denotes the absolute value. The function is given by ∣ x ∣ = { x x ≥ 0 − x x < 0 |x | = \begin{cases} x & x \geq 0 \\ -x & x < 0 \\ \end{cases} For example, ∣ 3 ∣ = 3 , ∣ − 2 ∣ = 2 |3| = 3, |-2| = 2 .


The answer is 2.

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Arron Kau by Arron Kau

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1 solution

Nahom Yemane
Jan 5, 2014

Sketching will show the given integral is equivalent to:

f ( x ) = 2 ∫ 0 π 2 s i n ( 2 x ) d x f(x)=2 \displaystyle\int^\frac{\pi}{2}_0 sin (2x) dx = 2 =2

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