Introduction of Triple Integration

We obtain volume of solid, G, by using triple intergrals in Cartesian Coordinate (x,y,z), as follow. [br] Volume, V = [math]\int\int\int[/math] dV =[math]\int\int\int[/math] dzdydx[br][br]Note that dz,dy,dx can be in any order
Type "Example 2.18" to see the graph in phone app[br][u][color=#0000ff]https://www.geogebra.org/3d/hkm6zzxm[/color][/u]
Example
Use triple integrals to find the volume of the solid which is bounded above by z=4-2x-y and below by region R in the xy-plane: [br][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWgAAABOCAYAAAANZ4xKAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAEnQAABJ0Ad5mH3gAABWVSURBVHhe7Z0JWFRVG8dfURYFFRRFRQXcUTP3pXJHyu1LzSU33Jc0QU0rNRMty6XcckvMJfc9y1JI0dRyQVPUMivAFRBBBREVTL/zP95rAw6b3Bnu0Pt7nnmYOffCnQvM/7zn3U6++4nxj4lhGIbRHVbKV4ZhGEZnsEAzDMPoFBZohmEYncICzTAMo1NYoBmGYXQKCzTDMIxOYYFmGIbRKSzQDMMwOoUFmmEYRqewQDMMw+gUFmiGYRidwgLNMAyjU1igGYZhdAoLNMMwjE5hgWYYhtEpLNAMwzA6hQWaYRhGp7BAMwzD6BQWaIZhGJ3CAs0wZmDlypXUtGlT+v333+mnn356+lxrYm/coA7tO9CQIUMo6d49eQ1cC9dnLA8WaIYxMRBJB3sHat6iBQUHB9ORI0fo0KFDVL16deUM7TgeEkKzP5stnyclJsprtG/fnho1aiTHGMuCBZphTAxEslv3btTGy4tCQ0PJ189POaI97dq1Izd3dypcuDDFCGsaVnTFChVNMhkwpocFmmHyGIUKFqRC9vZ0Qwj077/9RjVq1lCOMJYGCzTDmAH4ndeuXUuJiYkUcvw4/fDDD8oR01CxQgU6e/YsRYRHsPVswbBAa8D9+/fpmx07qG27tvTiiy9Sy5Yt6es1X9PDhw+VMywf3Mu0j6bRB5MnKyN5h5MnT9DgwYNlgM0U4Hf24969FBAQQBUrVaLZs2ZTwwYNlKOmoYIQ6E2bNrH1bOHku58Y/1h5zjwniJD/cf48TZ4yhQra2dGWLVvoiwULaOGiRVSnTh3lLMtmT2AgTfP3p1ZeXvTxRx8po5YNJtYlS5bQmjVrqHbtOvTFwi/IvlAh5ahlA4sdNG/eXH5lLBO2oDVgwIABNHPWLHKwt6f8+fNTzZovkEup0uTk5KScYdlEXouknw4coG7duysjlk9KSooU5zZtvGnUqFHi72ZF+fLlU45aNsgaQaYIi7PlwwKtIXADnDp1ipYuXUKj/fzIzc1NOWK54J6WfxVAvXr1IkdHR2XU8rG2tqYxY8aIyTTvuACQsYH857lz55o0U4QxHyzQGgGrBb7nQYMG0e34eCrhUjJPWGQ7tu+gsmXLCyGrqYwwegXZG/Bz44HnjOXDAp0NEEx64403ZCAQ1tc9YbGoIFKO4oP9+4OpfLlyNOrtURQW9rdyVN/A8po9e7YsZsADy2OA9x8Sclzm8BpONuf/+IO2bduuvNI3cbGx9N6771LdunWlO+Ovv/5SjmQd1TLF3z29h5aVgYhpGLuG4YMrA/8bcJAwi5w9e47mzplDU6dNpTNnztDOnTtp3vz5Ri2Vq1evCkt6IA0ZPJS6duuqjOoTuDBmzJxBpUuVoS5dOtMoX1/xvgdL/+XUqVNp+3bjQjx69Gjpe9czdxITaeLECdS+fQeq7ulJ77wzVtzTtGfSziB2v/zyC81fsMAklicENT1QuKIl5rwWY3pYoLPAP//8Qx988AGVKVNGBpQy4/r1GCFe/WnkyJGyzFbPnBEf2ilT/GUGQ9myZZVR40DIwsLDLSaLY/eePbRu3TpavGgRFSlSRBl9FlMLNMM8L+ziyAJxcbEy6R+5pcZAnqu//1S5FIZF+t1330qXQO3atZUz9MvJX38lewf7DAVMBUUWD8Q9YsLSO48fP5aumtIuLmRvb6+MGgf3ExUVRYl3EpQRhtEHLNBZ4Nq1SIqNjSUHBwdlJDWDBw2iR48fkVerVtSwYUM6dOgwffb55+Tq6qqcoU8gTCgFLiisxgLW1sqocSB2qITbFxxMmzdvVkb1y507dyg8LIxsxb0h9dEYWzZvoVdfe5UWL15MV65coa5du9PAgYPo5s2byhnaoXaVU33Iap6yKcE11K52jGXCAp0F8IE1DAimxd3dnaZNnUq/CBH7VVikq1evIs9q1ZSj+uX+gwcUd/OW8ipjmjRpQseOHZP317NnT2VUv9y7dz9ToUXwM3BPoLwn+GcPHvyJVqz4iooVK6acoQ2oUETq247t2+V1FixYICsLTQmuiQkVTZNyA6wqMRHVr1+fjh49poxqh+GEN8p3FD0Q/8umwDBA3NrLi8IjwpUj5sGsAp02Om0YiU5rYehp5r98+bK0wtKzxCyVe3fvUsz162SFIg1lLK8AtxSsaOsCBZSR3MO5RAmZ+oavAKuRXiae5DZt3kR9+vRRXmWda9eu0YdTptAUf39l5PmYKr6/bdu2VLx4MSpbVvuVJAK9Cxd+QUXEBFSlchWytbVVjmgLYhKfi9UwsoBKlypFzsWdlSPmwawCDcvLu00b+bxfv37k4+MjnwNPT08aO3aMfI5Zd8ann+omYAPfK/yYWltWuQ0s6JSHyVSiREnp5shLwKJCtWD58uWVkdzF0BILCwsjdw8P5Yj2wLWBatYSyoSQFVRh7tixI+385pscxxlgzMBt5upajpxM9LmxtrYhypfP5M2gcB8FxESP/yVzr0jMKtB2dnZUq/aTNKB69eqlskgRVLOxsZUiOG78eCrubN6ZKiOux8Qoz/IWCQkJlJSUN/2TuLeM3FLmBsYGrGi4OFyEJfbJJ58oR7QFro1z585mq8wbueGfCoOoW9eu9MMPuzWJneB3f+NGDFWtWsVk/U3gwrK1tSOPCqab7MD9pCS6LlaadevUlTplTszugy5fzrhFEx9/WyzLNtOAgQMswn/LMM8L3BtJd++axIX33a5dtGxZgLTUsWLdt28fNWncOMOgZOXKlWnhwoX0wgsvUIH8VmSVL+eyEBcXRxcjIkxagYq2CqVKuZjc7XDp8mXpLqtYqaIyYj7MLtAq4eGpne07v/2WbG1sqHPnLsqIPsBSD6lleRHVDZAXgVtKL8CqhXtDbWeKYCsa6pvChYfiIVjpeGzYsIFat25NR44eNWnjJHQF3Lhx49N2u6jYDFi+nJKTU8jNzV0564lVPWz4sKdxJsMHKlhPnDihnPksYeFhMhiI8+APRqEUctcN3Q74rG7ctJGavvIKtWjRQgq4imGM6+PpHyujz5IoJs758+fJtg04F5XDm8S9wQVYpsy/K4snRVAT5Xt5++235WuAbKDevXvJnW3gNsopZhdo5Num9XeipHjrlq301ltvUeF0UtkMwT86NsY0/ANn9MC56ocjuzxITpa9NfIienMDaEmMjtxSCA6+9NJLMgsA/48QFny4TQmsc2SOwIKGwJgKCNOHkyfTl19+SZMmTpIZMcuWfUlHjx4hZ+fiqQKE95KSKCY6RhZ7nTx5Uk4ia9eukzGQkW+PlG5PYxw8eJD69+svtKEwBQUFUkhIiKxJgOgauh3QauHB/Qe0dt06srWzpfXiq+pLh596/LhxsklWyxYt5VhaUGA2fNgw8TsLpsVLltDp06fpvffekz8X1rOjY1F5Hn7miq+WU48e3Wne3Hny/YSKc6Mio2jOnDk0Y8ZMuSGDFq4isws0oq34JamgsGPZl8vk7JbV3sn4h9/1/a6nlkJmD5yrRtAZJjcwtGzN0czI0OeNa5uKHTu200EhYJMmTaJXhOWKuBJ87HBlugvr2TCoFhkVTW7ubtSrd28ZdEP7BF8/X+rTpzf17dPXqH8Xojl//nyqXqM6TRCTWtGijvIamOhg6Bm6HWA1I/kAFbG1ar5AFy9dlK4JFVj01WvUoFrie9MC0V2+PEBavdOnfyLdrHg/FTw8yMmpWKpMEVzfz2+0eA+1qZpnVRmMPX78OM2bN1fm0ZcrV06epwVmF2grcXNWVlZPZ7bDhw/TlatXyMenn9E/EMMw+uRmXBzt/GantHwhzip3Eu5IoYMYGiYCoLXrvHnz5OQBV4C//xTq2KGDTAdM77O//0AwXbp0iXz6+qRaXSOw6ezsLKzUMsrIv+CaVYXAxsTcoOjr1+UY9Obkryepn0/qn6MSERFBe/fuJW9v71QtaKPFBJEgRB7+eWNgwkCQcv369dSgQUNxnrY+d7MLtHOxYnKpclH80hGFDQhYRj5i+eLiUlI5g2EYS+DPv/6W4gmXATK0VCB2t2/dIk9P4+lvcAX4+frRi7VfJF9fX2lNGwMxkp9//pk8PNyphhB7FYyHngmVwgiBNEY1IdCIQ6iuzQsXLkjD8OWXX5av03L61Glpbbds2SrVZIGdkqxtrKm8m/HkBljVFTwqUMmSJalps2bKqHbkWpCQHj+mDRvWU1nXsuTVurUymDXM6YNmGMY4f/39pwwylyj5r/sQlmpg4B4qXKSw0QIVuCzGvzuePKt70rhx49MVZ3BXCOyli5fI0dGJbAwKUZBHDkHNqECljLCsHZ2cKOJihHxPu3fvoTfffDPVRGLIb78/aXlQRLxvFfjxA4MCMyxQQfxs3959FH/7NkVG5jwomBazCzR+0Y5OjjLCum9vMA0dNjTDP5Ix2AfNMPoELktkYxgrUJEBxSmT5c4870+YIF0d0dHXZRYE3CVZAT8DJeyYGDIqUIGgQljDw8Ip5HiI9IVnJ+UPzbbQ9Cwi4mK6BSp4LytXraaJkyaSQ2EHGVTUGrMLNKpyMFOhe1inLp2oYsVKyhF9g+UR/Od5ET2UQ5uKAtZ5995MQVR0NMUnxFNUZOTT1LH0qFypigz4X7jwpwz2BwcH0+Url2VvGgBfNApgzp8/L3/WhPffl+Ofzpgh/cCwUJcuXUopD1OooJFiFntxDoKKKBLBChguUWRP1KtXX14PInr0yFGZraLGtFRQ+QuBDj0dStu3b6MePVJvOpGWGtVryIwmuGySk5Np85bNZCPuzcnpSeYGLH/0R8eEApBauHDBF9SieXOqXaeOEHE3GSi8LSxpvB/csxbkmosDGRt6y3nOCPjN4T/Pi+ilHNoUpFcYxfwLhOl9IZ7o7IcsCKRfwgp+tU0b2c8cXf+MUb9+Perbt4/cg7Nt23bSN9yrZy9q3LixsCZP0ciRI8jbu41s44BMLaSrQVBfefll6XpEAQ2yQODDTZt6C+C+GDFipDSOOnXqRDOFsPfo0VMGJCtVqiQzRw4dPkTDhg9PFYwEaqDw1u1b1NenX7q+apVX275GXl5e8meiwMfD3YM6dvwf1a1bjwIDA2W6Xf/+/alECWeaIKz/1157jZo2ayq/BysBuE+QatetW3dq0KCBvGct4Ib9mYBZ3s/XVwY2Vq1amadcJaguQ5DGEnZHyS5oxIWMAXSO492t/3vAug4IWE52BW1lBoilkntBQgujpIsLFcyk8bupwB6A6l6IHTp0kEspLcktCxofIvSWRuUXKrJQwIA9BLUCnc6Km7n7GMDyGxODusfjtGlPNnNgzAcKdG7ejJMWvSXDAp0JD1NS5BLQ3qEQFcgFHzR8Xx9O/kAsnbrJ6iu4hfz9/WUeaU5BGhJ8iPb2mVdvmgJY8KtXr6ZlAQF0YP9+Sk5+QLNmzZICl1NwbzZ2dlTI3vxd+tZvWE/Hjh2VO6Jv3bqVTp8KpZUrvpITEmMa4F6B6+HsmTP0zY4d8rOClWF2ExD0Bgt0JiSLP/ztW7dlVDi9lB5TcuJEiPwKnxf+2Tp2bC99csc0aIKOcmgEU5BVY24Q1Pl+1y5q1qyZ9NcVKVqUegpr5+Svp2SgKaegAyG6qDkWeRLkMRfw3wYFBlHnTl1kqheqyjr+ryMdOHCA4vNoywA9cEf83s+eOUsjRoyghMQ79M4776SbUmdJsEBnEfSoVkFEGqlBaKgC9wMsvunTpwsrt2uW04WyCoI1xYoXfzo5OBQuQqVLl5Z5m1oAASkjfh5Q7wtl92qjGbXJDIIg6DioFcgbvfDnn1LEVJDsn5KSTJHXIpWRnFG1ShUq6vhk8oFwomLNMD8e7ofdu3fL41oRGRUlq+hKlS6ljJDsG2FY1cZoj5p6e+jwYelztnTLWYUFOhNQ6okdR5BYr7L3xyAaPvwtqlOnLh0+dIg2bNwgP+xbtmyVYpqWtDvJGHsY7i5jSIoQf8OG+uj45yisTYznBCy3US6LElZ1w9hd331HQ4cMoVq1alFQUJAcQ64pIvstWjR/JhIOv6rahD69B8QdIp+WR48e0aN/HqXKssD7sLOzfSZlKrvcTUqSIt+oUeOn0f3YuFgq6+pK+/fvl/mqY8eOldYWViZpyUohVHrFT4/Eey8kLHfDzR1wX7a2NvIYw2QHFmgjfP/99zRo0CBKEEvS4OB9UmSQdqMCPzDq9V9q0oTWrFkr++eirWN6GDbKSe+R1SwKCA42Qn0eYEXiOnsCA+nixYt0+fIl6tr1DeXokx1v0EgGKZDYew2+d4glevu2F4KUFsOGPOk9kFqV1R0v7MQqwbqAjfIqe1y9elXcS1e5+/pv587J992qdSvlqLBiPSrQjJkzycnJidatWycnqPR6QGSlECo7xU9Y/cjdPxgmm7BAGyHxbiKdEx/yNt7eMvgwUiz7jX2QPSq4U3FhKXl5tTF6XCuuR0c/zQLAV7zOSlvWtMAV8+DBfXrv3Xdp3LhxNGbMGKOFQtguKfZGrGwPib4KcEOgQ5gpMOwLHiMs0rv3koSgZd93iMkkSbxfWPuLFi2WmRNptyiDKAeKlQHEHOKcNndWKzAR3jCwrvEc24vZFbR8nyhjXligjdCjew/ZVB2P8ePHS0sxLfDXbtywkW4nxGdag58TFwe2DELVZeKdBPka2++goup5dqqAYK1fv0FagNu2bZMVWcZADwUrq3x0LTJKWsDt27U3OgHlxMWBKjHXcq4UISx5Nbvh1q1bZF+wkNEeDpmBXUHQg1fdVd3NzU058i9IvTp96pScnDLyUebExVHKxUVa6YYTT1R0VK5sOMpYPlyo8hygzBMlqqgkWrVqFVWrWpVq1KhJdnYFqXHjRspZ2oBmLNjIABVU/fr5CPH5mn78MYgWL15ssk1s4cMdO2Y0uQqruVXLVqlaSWoJKtQWL11C8+bOkS4kNNGpVKmyFFCtVyTY+v/HvUE0evQYOeGuWrVa+rt79+6tnKENmGw+++wzOvLLEZq/YL4cQ7rXmz3epG7du8nXDJNV2ILOJmh/6O39qgyu1atbj1o0bylFMzT0tCx91Rq4ID78cIrcUqhhw4YUGLhbbolvKnEGEDAUeOS3yk9NmjRRRrWnc5fO1LlTJxo6dJh0Jzk7l6DhYjLSWpzRK2LCxAly1x6UGaMoBm1uTbEbNN477qGaZzV6/fXXxWqsu2wkj3tlmOzCFjTzDChrX7FyBfn6+T2Xr5thGG1gC5pJBbp1LRPW5cABA1mcGSaXYQuakaDABjsuFyrkIDMgjAXZGIYxLyzQDMMwOoVdHAzDMDqFBZphGEansEAzDMPoFBZohmEYncICzTAMo1NYoBmGYXQKCzTDMIxOYYFmGIbRKSzQDMMwOoUFmmEYRqewQDMMw+gUFmiGYRidwgLNMAyjU1igGYZhdAnR/wE4iVlbbzbIrQAAAABJRU5ErkJggg==[/img]
Cerrar

Información: Introduction of Triple Integration