91影视

We have been given the triple integral:

$\begin{array}{r}{鈭�}_{\mathcal{R}}鈥�\ln 鈦�\left(xy{z}^2\right)dV\text{, where}\\ \mathcal{R}=\left\{\right(x,y,z)鈭�1鈮�x鈮�3,1鈮�y鈮�e,\text{and}1鈮�z鈮�2\}\end{array}$

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

By Fubini's theorem of triple integral :

$\begin{array}{r}{鈭�}_R鈥�\mathrm{In}鈦�\left(xy{z}^2\right)dv={鈭�}_1^3鈥�{鈭�}_1^e鈥�{鈭�}_1^2鈥�\mathrm{In}鈦�\left(xy{z}^2\right)dzdydx\\ ={鈭�}_1^3鈥�{鈭�}_1^e鈥�{鈭�}_1^2鈥�(\mathrm{In}鈦�x+\mathrm{In}鈦�y+2\mathrm{In}鈦�z)dzdydx\\ ={鈭�}_1^3鈥�{鈭�}_1^e鈥�\left[{鈭�}_1^2鈥�(\mathrm{In}鈦�x+\mathrm{In}鈦�y+2\mathrm{In}鈦�z)dz\right]dydx\\ ={鈭�}_1^3鈥�{鈭�}_1^e鈥�\left[\mathrm{In}鈦�x{鈭�}_1^2鈥�dz+\mathrm{In}鈦�y{鈭�}_1^2鈥�dz+2{鈭�}_1^2鈥�\mathrm{In}鈦�zdz\right]dydx\\ ={鈭�}_1^3鈥�{鈭�}_1^e鈥�\left[\mathrm{In}鈦�x(z{)}_1^2+\mathrm{In}鈦�y(z{)}_1^2+2(z\mathrm{In}鈦�z鈭�z{)}_1^2\right]dydx\\ ={鈭�}_1^2鈥�{鈭�}_1^e鈥�[\mathrm{In}鈦�x(2鈭�1)+\mathrm{In}鈦�y(2鈭�1)+2(2\mathrm{In}鈦�2鈭�2鈭�1\ln 鈦�1+1\left)\right]dydx\end{array}$

Integrate with respect to y

$\begin{array}{r}={鈭�}_1^2鈥�\left[{鈭�}_1^e鈥�(\mathrm{In}鈦�x+\mathrm{In}鈦�y+2(2\mathrm{In}鈦�2鈭�1\left)\right]dy\right]dx\\ ={鈭�}_1^2鈥�\left[\mathrm{In}鈦�x{鈭�}_1^e鈥�dy+{鈭�}_1^e鈥�\mathrm{In}鈦�ydy+2(2\mathrm{In}鈦�2鈭�1){鈭�}_1^e鈥�dy\right]dx\\ ={鈭�}_1^2鈥�\left[\mathrm{In}鈦�x(y{)}_1^e+(y\mathrm{In}鈦�y鈭�y{)}_1^e+2(2\mathrm{In}鈦�2鈭�1)(y{)}_1^e\right]dx\\ ={鈭�}_1^2鈥�[\mathrm{In}鈦�x(e鈭�1)+(e\mathrm{In}鈦�e鈭�e+1)+2(2\mathrm{In}鈦�2鈭�1\left)\right(e鈭�1\left)\right]dx\end{array}$

Since $\ln e=1,\ln 1=0$

localid="1650348241491" $\begin{array}{r}=(e鈭�1){鈭�}_1^3鈥�\mathrm{In}鈦�xdx+{鈭�}_1^3鈥�dx+2(2\mathrm{In}鈦�2鈭�1)(e鈭�1){鈭�}_1^3鈥�dx\\ =(e鈭�1)[x\mathrm{In}鈦�x鈭�x{]}_1^3+(x{)}_1^3+2(2\mathrm{In}鈦�2鈭�1)(e鈭�1)[x{]}_1^3\\ =(e鈭�1)[3\mathrm{In}鈦�3鈭�3+1]+(3鈭�1)+2(2\mathrm{In}鈦�2鈭�1)(e鈭�1)(3鈭�1)\\ =(e鈭�1)[3\mathrm{In}鈦�3鈭�2]+2+4(2\mathrm{In}鈦�2鈭�1)(e鈭�1)\\ =(e鈭�1)[3\mathrm{In}鈦�3鈭�2+8\mathrm{In}鈦�2鈭�4]+2\\ =(e鈭�1)[3\mathrm{In}鈦�3+8\mathrm{In}鈦�2鈭�6]+2\end{array}$