Question

Evaluate the iterated integral :

${\int }_0^1 {\int }_0^{6-y} {\int }_0^{3-3y-(1/2)z} (y-z)dxdzdy$

Short answer

$\frac{145}{48}$

Step-by-step solution

Step 1. Given information.

Given integral is :

${\int }_0^1 {\int }_0^{6-y} {\int }_0^{3-3y-(1/2)z} (y-z)dxdzdy$

We have to evaluate it.

Step 2. Evaluate.

Given ${\int }_0^1 {\int }_0^{6-y} {\int }_0^{3-3y-(1/2)z} (y-z)dxdzdy$

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

Step 3. Integrate with respect to z.

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

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