definite integral

**ice_syncer** · February 20th 2011, 05:07 AM

what is the value of

integral ( from 0 to 2pi ) |arcsin(sinx)| dx

where | | means absolute value
..answer is pi*pi/2.0 ;

**tonio** · February 20th 2011, 05:26 AM

Originally Posted by ice_syncer

what is the value of

integral ( from 0 to 2pi ) |arcsin(sinx)| dx

where | | means absolute value
..answer is pi*pi/2.0 ;

As $\arcsin(\sin(x))=x$ , then...

Tonio

**Opalg** · February 20th 2011, 08:38 AM

Originally Posted by ice_syncer

what is the value of

integral ( from 0 to 2pi ) |arcsin(sinx)| dx

where | | means absolute value
..answer is pi*pi/2.0 ;

$\arcsin t$ means the value of $\theta$ between $-\pi/2$ and $\pi/2$ such that $\sin\theta = t$ . So $\arcsin(\sin x) = x$ when $0\leqslant x\leqslant\pi/2.$

When $\pi/2\leqslant x\leqslant 3\pi/2$ you should use the fact that $\sin x = \sin(\pi-x)$ to deduce that $\arcsin (\sin x) = \pi-x$ .

For $3\pi/2\leqslant x\leqslant 2\pi$ , $\arcsin (\sin x)$ will be given by yet another formula, which I'll leave you to think about.

Once you have the formulas for $\arcsin (\sin x)$ on the three different intervals, you should find it comparatively easy to take their absolute values and integrate over the corresponding intervals.

When you have done this, it would be a good idea to check your answer by drawing the graph of $\left|\arcsin(\sin x)\right|$ on the interval from $0$ to $2\pi$ . It should consist of straight line segments, and the area under the graph will consist of a number of triangles. You can then check that your value for the integral is correct by calculating the total area of these triangles using the formula "half base times height".

**ice_syncer** · February 20th 2011, 09:00 AM

no... arcsin is only defined for -pi/2 to pi/2

**Opalg** · February 20th 2011, 10:24 AM

Originally Posted by ice_syncer

no... arcsin is only defined for -pi/2 to pi/2

No, arcsin is defined on the interval –1 to 1. The values of the arcsin function lie in the interval $-\pi/2$ to $\pi/2.$

Since sin x always lies in the interval –1 to 1, it follows that arcsin(sin x) is defined for all x, and its values always lie in the interval $-\pi/2$ to $\pi/2.$

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