What is the integral from 0 to pi/2 of | 1/2-sin(x)| ?

The integral from 0 to pi/2 of | 1/2-sin(x)| is sqrt(3)-pi/(12)-1

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integral from 0 to pi/2 of | 1/2-sin(x)|

\int_{0}^{\frac{π}{2}} \frac{1}{2} - sin (x) d x

3 - \frac{π}{12} - 1

Decimal

0.47025 \dots

Solution steps

\int_{0}^{\frac{π}{2}} \frac{1}{2} - sin (x) d x

Eliminate Absolutes

= \int_{0}^{\frac{π}{6}} \frac{1}{2} - sin (x) d x + \int_{\frac{π}{6}}^{\frac{π}{2}} - \frac{1}{2} + sin (x) d x

\int_{0}^{\frac{π}{6}} \frac{1}{2} - sin (x) d x = \frac{π}{12} + \frac{3}{2} - 1

\int_{\frac{π}{6}}^{\frac{π}{2}} - \frac{1}{2} + sin (x) d x = - \frac{π}{6} + \frac{3}{2}

= \frac{π}{12} + \frac{3}{2} - 1 - \frac{π}{6} + \frac{3}{2}

Simplify

\frac{π}{12} + \frac{3}{2} - 1 - \frac{π}{6} + \frac{3}{2} : 3 - \frac{π}{12} - 1

= 3 - \frac{π}{12} - 1

What is the integral from 0 to pi/2 of | 1/2-sin(x)| ?
The integral from 0 to pi/2 of | 1/2-sin(x)| is sqrt(3)-pi/(12)-1