Mathematics-I (3110014)

$\mathrm{Let}, \mathrm{x}=\mathrm{r} \mathrm{sin\theta } \mathrm{cos\varphi } ; \mathrm{y}=\mathrm{r} \mathrm{sin\theta } \mathrm{sin\varphi } ; \mathrm{z}=\mathrm{r} \mathrm{cos\theta } .$

$\mathrm{Jacobian}, \mathrm{J}=\left|\begin{array}{ccc}\frac{\partial \mathrm{x}}{\partial \mathrm{r}}& \frac{\partial \mathrm{x}}{\partial \mathrm{\theta }}& \frac{\partial \mathrm{x}}{\partial \mathrm{\varphi }}\\ \frac{\partial \mathrm{y}}{\partial \mathrm{r}}& \frac{\partial \mathrm{y}}{\partial \mathrm{\theta }}& \frac{\partial \mathrm{y}}{\partial \mathrm{\varphi }}\\ \frac{\partial \mathrm{z}}{\partial \mathrm{r}}& \frac{\partial \mathrm{z}}{\partial \mathrm{\theta }}& \frac{\partial \mathrm{z}}{\partial \mathrm{\varphi }}\end{array}\right|$

$\mathrm{Jacobian}, \mathrm{J}=\left|\begin{array}{ccc}\mathrm{sin\theta } \mathrm{cos\varphi }& \mathrm{r} \mathrm{cos\theta } \mathrm{cos\varphi }& -\mathrm{r} \mathrm{sin\theta } \mathrm{sin\varphi }\\ \mathrm{sin\theta } \mathrm{sin\varphi }& \mathrm{r} \mathrm{cos\theta } \mathrm{sin\varphi }& \mathrm{r} \mathrm{sin\theta } \mathrm{cos\varphi }\\ \mathrm{cos\theta }& -\mathrm{r} \mathrm{sin\theta }& 0\end{array}\right|$

$\mathrm{Jacobian}, \mathrm{J}=\mathrm{sin\theta } \mathrm{cos\varphi } \left(0+{\mathrm{r}}^2 {\sin }^2\mathrm{\theta } \mathrm{cos\varphi }\right) - \mathrm{r} \mathrm{cos\theta } \mathrm{cos\varphi } (0-\mathrm{r} \mathrm{sin\theta } \mathrm{cos\theta } \mathrm{cos\varphi })- \mathrm{r} \mathrm{sin\theta } \mathrm{sin\varphi } (-\mathrm{r} {\sin }^2\mathrm{\theta } \mathrm{sin\varphi } - \mathrm{r} {\cos }^2\mathrm{\theta } \mathrm{sin\varphi })$

$\mathrm{Jacobian}, \mathrm{J}={\mathrm{r}}^2 {\sin }^3\mathrm{\theta } {\cos }^2\mathrm{\varphi }+{\mathrm{r}}^2 \mathrm{sin\theta } {\cos }^2\mathrm{\theta } {\cos }^2\mathrm{\varphi }+{\mathrm{r}}^2 {\sin }^3\mathrm{\theta } {\sin }^2\mathrm{\varphi }+{\mathrm{r}}^2 \mathrm{sin\theta } {\cos }^2\mathrm{\theta } {\sin }^2\mathrm{\varphi }$

$\mathrm{Jacobian}, \mathrm{J}={\mathrm{r}}^2 {\sin }^3\mathrm{\theta } {\cos }^2\mathrm{\varphi }+{\mathrm{r}}^2 {\sin }^3\mathrm{\theta } {\sin }^2\mathrm{\varphi }+{\mathrm{r}}^2 \mathrm{sin\theta } {\cos }^2\mathrm{\theta } {\cos }^2\mathrm{\varphi }+{\mathrm{r}}^2 \mathrm{sin\theta } {\cos }^2\mathrm{\theta } {\sin }^2\mathrm{\varphi }$

$\mathrm{Jacobian}, \mathrm{J}={\mathrm{r}}^2 {\sin }^3\mathrm{\theta } \left({\cos }^2\mathrm{\varphi }+{\sin }^2\mathrm{\varphi }\right) +{\mathrm{r}}^2 \mathrm{sin\theta } {\cos }^2\mathrm{\theta } \left({\cos }^2\mathrm{\varphi }+ {\sin }^2\mathrm{\varphi }\right)$

$\mathrm{Jacobian}, \mathrm{J}={\mathrm{r}}^2 {\sin }^3\mathrm{\theta }+{\mathrm{r}}^2 \mathrm{sin\theta } {\cos }^2\mathrm{\theta }$

$\mathrm{Jacobian}, \mathrm{J}={\mathrm{r}}^2 \mathrm{sin\theta } \left({\sin }^2\mathrm{\theta } + {\cos }^2\mathrm{\theta } \right)$

$\mathrm{Jacobian}, \mathrm{J}={\mathrm{r}}^2 \mathrm{sin\theta }$

$\mathrm{Now},\iiint \mathrm{xyz} \mathrm{dx} \mathrm{dy} \mathrm{dz}$

$N,={\int }_0^{\frac{\mathrm{\pi }}{2}}{\int }_0^{\frac{\mathrm{\pi }}{2}}{\int }_0^2 \left(\mathrm{r} \mathrm{sin\theta } \mathrm{cos\varphi }\right) \left(\mathrm{r} \mathrm{sin\theta } \mathrm{sin\varphi }\right) \left(\mathrm{r} \mathrm{cos\theta }\right) \left({\mathrm{r}}^2 \mathrm{sin\theta }\right) d\mathrm{r} d\mathrm{\theta } \mathrm{d\varphi }$

$N,={\int }_0^{\frac{\mathrm{\pi }}{2}}{\int }_0^{\frac{\mathrm{\pi }}{2}}{\int }_0^2 \left({\mathrm{r}}^5\right)\left({\sin }^3\mathrm{\theta } \mathrm{cos\theta }\right) \left(\mathrm{cos\varphi } \mathrm{sin\varphi }\right) d\mathrm{r} d\mathrm{\theta } \mathrm{d\varphi }$

$N,=\left({\int }_0^{\frac{\mathrm{\pi }}{2}}\left(\mathrm{cos\varphi } \mathrm{sin\varphi }\right) \mathrm{d\varphi }\right)\left({\int }_0^{\frac{\mathrm{\pi }}{2}}\left({\sin }^3\mathrm{\theta } \mathrm{cos\theta }\right) d\mathrm{\theta }\right)\left({\int }_0^2 \left({\mathrm{r}}^5\right) d\mathrm{r}\right)$

$N,=\left({\int }_0^{\frac{\mathrm{\pi }}{2}}\left(\frac{\sin 2\mathrm{\varphi }}{2}\right) \mathrm{d\varphi }\right)\left({\int }_0^{\frac{\mathrm{\pi }}{2}}\left({\sin }^3\mathrm{\theta } \mathrm{cos\theta }\right) d\mathrm{\theta }\right)\left({\int }_0^2 \left({\mathrm{r}}^5\right) d\mathrm{r}\right)$

$N,={\left[-\frac{\cos 2\mathrm{\varphi }}{2}\right]}_0^{\frac{\mathrm{\pi }}{2}} {\left[\frac{{\sin }^4\mathrm{\theta }}{4}\right]}_0^{\frac{\mathrm{\pi }}{2}} {\left[\frac{{\mathrm{r}}^6}{6}\right]}_0^2$

$N,=\left[-\frac{\cos 2\left(\frac{\mathrm{\pi }}{2}\right)-\cos 0}{2}\right] \left[\frac{{\sin }^4\frac{\mathrm{\pi }}{2}-{\sin }^{4}0}{4}\right] \left[\frac{2^6-0^6}{6}\right]$

$N,=\left[-\frac{-1-1}{4}\right] \left[\frac{1-0}{4}\right] \left[\frac{64-0}{6}\right]$

$N,=\frac{2}{4}· \frac{1}{4}·\frac{64}{6}$

$N,=\frac{4}{3}$

$\mathrm{Hence}, \iiint \mathrm{xyz} \mathrm{dx} \mathrm{dy} \mathrm{dz}=\frac{4}{3}$