[tex]\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{\varphi=\pi/3}^{\varphi=\pi/2}\int_{\rho=0}^{\rho=2}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta[/tex]
[tex]=\displaystyle\left(\int_{\theta=0}^{\theta=2\pi}\mathrm d\theta\right)\left(\int_{\varphi=\pi/3}^{\varphi=\pi/2}\sin\varphi\,\mathrm d\varphi\right)\left(\int_{\rho=0}^{\rho=2}\rho^2\,\mathrm d\rho\right)[/tex]
[tex]=\displaystyle2\pi\left(-\cos\varphi\bigg|_{\varphi=\pi/3}^{\varphi=\pi/2}\right)\left(\frac13\rho^3\bigg|_{\rho=0}^{\rho=2}\right)[/tex]
[tex]=\dfrac{8\pi}3[/tex]
