91影视

We have been given the triple integral:

$\begin{array}{r}{鈭�}_{\mathcal{R}}鈥�z\sin 鈦�x\cos 鈦�ydV,\text{where}\\ \mathcal{R}=\left\{(x,y,z)鈭�0鈮�x鈮�蟺,\frac{3蟺}{2}鈮�y鈮�2蟺,\text{and}1鈮�z鈮�3\right\}\end{array}$

We have to evaluate this over the specified rectangular solid regions.

By Fubini's theorem of triple integral :

$\begin{array}{r}{鈭�}_R鈥�z\sin 鈦�x\cos 鈦�ydV={鈭�}_0^{蟺/2蟺}鈥�{鈭�}_{\frac{3蟺}{2}}^3鈥�{鈭�}_1^3鈥�z\sin 鈦�x\cos dxdydz\\ ={鈭�}_0^{蟺}鈥�{鈭�}_{\frac{3蟺}{2}}^{2蟺}鈥�\left[{鈭�}_1^3鈥�z\sin 鈦�x\cos 鈦�ydz\right]dydx\\ ={鈭�}_0^{蟺}鈥�{鈭�}_{\frac{3蟺}{2}}^{2蟺}鈥�\left[\sin 鈦�x\cos 鈦�y{鈭�}_1^3鈥�zdz\right]dydx\\ ={鈭�}_0^{蟺}鈥�{鈭�}_{\frac{3蟺}{2}}^{2蟺}鈥�\left[\sin 鈦�x\cos 鈦�y{\left[\frac{z^2}{2}\right]}_1^3\right]dydx\\ ={鈭�}_0^{蟺}鈥�{鈭�}_{\frac{3蟺}{2}}^{2蟺}鈥�\left[\sin 鈦�x\cos 鈦�y\left[\frac{3^2}{2}鈭�\frac{1^2}{2}\right]dydx\right.\\ ={鈭�}_0^{蟺}鈥�{鈭�}_{\frac{3蟺}{2}}^{2蟺}鈥�[4\sin 鈦�x\cos 鈦�y]dydx\end{array}$

Integrate with respect to y

$\begin{array}{r}={鈭�}_0^{蟺}鈥�\left[4\sin 鈦�x{鈭�}_{\frac{3蟺}{2}}^{2蟺}鈥�[\cos 鈦�y]dy\right]dx\\ ={鈭�}_0^{蟺}鈥�\left[4\sin 鈦�x[\sin 鈦�y{]}_{\frac{3蟺}{2}}^{2蟺}\right]dx\\ ={鈭�}_0^{蟺}鈥�\left[4\sin 鈦�x\left[\sin 鈦�2蟺鈭�\sin 鈦�\frac{3蟺}{2}\right]\right]dx\\ ={鈭�}_0^{蟺}鈥�[4\sin 鈦�x[0鈭�(鈭�1)\left]\right]dx\\ ={鈭�}_0^{蟺}鈥�[4\sin 鈦�x]dx\\ =4\left[{鈭�}_0^{蟺}鈥�(\sin 鈦�x)\right]\\ =4\left[\right(鈭�\cos 鈦�x){]}_0^{蟺}\\ =鈭�4\left[\right(\cos 鈦�蟺)鈭�(\cos 鈦�0\left)\right]\\ =鈭�4[鈭�1鈭�1]\\ =鈭�4[鈭�2]\\ =8\end{array}$