91Ó°ÊÓ

Given integral is :

${∫}_0^1 {∫}_0^{6−y} {∫}_0^{3−3y−(1/2)z} (y−z)dxdzdy$

We have to evaluate it.

Given ${∫}_0^1 {∫}_0^{6−y} {∫}_0^{3−3y−(1/2)z} (y−z)dxdzdy$

$\begin{array}{r}={∫}_0^1 {∫}_0^{6−y} \left[{∫}_0^{3−3y−\frac{1}{2}z} (y−z)dx\right]dzdy\\ ={∫}_0^1 {∫}_0^{6−y} \left[(y−z){∫}_0^{3−3y−\frac{1}{2}z} dx\right]dzdy\\ ={∫}_0^1 {∫}_0^{6−y} \left[(y−z)(x{)}_0^{3−3y−\frac{1}{2}z}\right]dzdy\\ ={∫}_0^1 {∫}_0^{6−y} \left[(y−z)\left(3−3y−\frac{1}{2}z\right)\right]dzdy\\ ={∫}_0^1 {∫}_0^{6−y} \left[y\left(3−3y−\frac{1}{2}z\right)−z\left(3−3y−\frac{1}{2}z\right)\right]dzdy\\ ={∫}_0^1 {∫}_0^{6−y} \left[3y−3y^2−\frac{yz}{2}−3z+3zy+\frac{z^2}{2}\right]dzdy\\ ={∫}_0^1 {∫}_0^{6−y} \left[3y−3y^2+\frac{5yz}{2}−3z+\frac{z^2}{2}\right]dzdy\end{array}$

$$\begin{gathered} \begin{array}{r}={∫}_0^1 \left[3y{∫}_0^{6−y} dz−3y^2{∫}_0^{6−y} dz+\frac{5y}{2}{∫}_0^{6−y} zdz−3{∫}_0^{6−y} zdz+\frac{1}{2}{∫}_0^{6−y} z^{2}dz\right]dy\\ ={∫}_0^1 \left[3y(z{)}_0^{6−y}−3y^2(z{)}_0^{6−y}+\frac{5y}{2}{\left(\frac{z^2}{2}\right)}_0^{6−y}−3{\left(\frac{z^2}{2}\right)}_0^{6−y}+\frac{1}{2}{\left(\frac{z^3}{3}\right)}_0^{6−y}\right]dy\\ ={∫}_0^1 \left[3y(6−y)−3y^2(6−y)+\frac{5y}{2}\left(\frac{(6−y{)}^2}{2}\right)−3\left(\frac{(6−y{)}^2}{2}\right)+\frac{1}{2}\left(\frac{(6−y{)}^3}{3}\right)\right]dy\\ ={∫}_0^1 (6−y)\left[3y−3y^2+\frac{5y(6−y)}{4}−3\left(\frac{(6−y)}{2}\right)+\frac{1}{2}\left(\frac{(6−y{)}^2}{3}\right)\right]dy\\ ={∫}_0^1 (6−y)\left[3y−3y^2+\frac{\left(30y−5y^2\right)}{4}−\left(\frac{18−3y}{2}\right)+\left(\frac{36−12y+y^2}{6}\right)\right]dy\\ ={∫}_0^1 (6−y)\left[\left(\frac{36y−36y^2+90y−15y^2−108+18y+72−24y+2y^2}{12}\right)\right]dy\\ ={∫}_0^1 (6−y)\left[\left(\frac{−49y^2+120y−36}{12}\right]dy\right.\end{array} \\ \begin{array}{r}=\frac{1}{12}{∫}_0^1 \left(49y^3−414y^2+756y−216\right)dy\\ =\frac{1}{12}\left[−414{∫}_0^1 \left(y^2\right)dy−216{∫}_0^1 dy+49{∫}_0^1 \left(y^3\right)dy+756{∫}_0^1 \left(y\right)dy\right]\\ =\frac{1}{12}\left[−414{\left[\frac{y^3}{3}\right]}_0^1−216[y{]}_0^1+49{\left[\frac{y^4}{4}\right]}_0^1+756{\left[\frac{y^2}{2}\right]}_0^1\right]\text{Integ}\\ =\frac{1}{12}\left[−414\left[\frac{1}{3}\right]−216\left[1\right]+49\left[\frac{1}{4}\right]+756\left[\frac{1}{2}\right]\right]\\ =\frac{1}{12}\left[−138−216+\left[\frac{49}{4}\right]+378\right]\\ =\frac{1}{12}{\left[24+\left[\frac{49}{4}\right]\right]}_0\left[\frac{1}{12}\left[\frac{96+49}{4}\right]\right.\\ =\frac{145}{48}\end{array} \end{gathered}$$uncaught exception: Invalid chunk

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Invalid chunk') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Invalid chunk') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Invalid chunk') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Invalid chunk') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Invalid chunk') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('49135e911ee0fd7...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('

$\begin{array}{r}=\frac{1}{12}{∫}_0^1 \left(49y^3−414y^2+756y−216\right)dy\\ =\frac{1}{12}\left[−414{∫}_0^1 \left(y^2\right)dy−216{∫}_0^1 dy+49{∫}_0^1 \left(y^3\right)dy+756{∫}_0^1 \left(y\right)dy\right]\\ =\frac{1}{12}\left[−414{\left[\frac{y^3}{3}\right]}_0^1−216[y{]}_0^1+49{\left[\frac{y^4}{4}\right]}_0^1+756{\left[\frac{y^2}{2}\right]}_0^1\right]\text{Integ}\\ =\frac{1}{12}\left[−414\left[\frac{1}{3}\right]−216\left[1\right]+49\left[\frac{1}{4}\right]+756\left[\frac{1}{2}\right]\right]\\ =\frac{1}{12}\left[−138−216+\left[\frac{49}{4}\right]+378\right]\\ \left.=\frac{1}{12}{\left[24+\left[\frac{49}{4}\right]\right]}^1\right]\\ =\frac{1}{12}\left[\frac{96+49}{4}\right]\\ =\frac{145}{48}\end{array}$