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Definite integral

\begin{align} \int_{ 0 }^{ \pi } \sin ^3\left(x\right) \,\mathrm{d}x&= \left[ {{\cos ^3\left(x\right)}\over{3}}-\cos \left(x\right) \right]_{ 0 }^{ \pi }\\ &= {{2}\over{3}}-\left(-{{2}\over{3}}\right) \\&= {{4}\over{3}} \\& \approx 1.333333333333333 \end{align} Help to find the primitive function

Mean value

\begin{align} \mu&=\frac{1}{b-a}\int_{a}^{b}f(x)\,\mathrm{d}x\\&=\frac{ {{4}\over{3}} }{ \pi }\\&= {{4}\over{3\,\pi}} \\&\approx 0.42441318157839 \end{align}

Graph of the function and mean value