Answer to Question #221375 in Calculus for Sree

Question #221375
Find the volume of the region in the first octant bounded by coordinate planes and the planes x+z=1,y+2z=2
1
Expert's answer
2021-08-02T07:12:45-0400

"z= 1-x; z= \\frac{2-y}{2}\\\\\n1-x = \\frac{2-y}{2}\\\\\n2-2x=2-y\\\\\ny=2x\\\\\nz= 0 \\implies y= 2\\\\\nVolume = \\int\\int_D\\int _{z=0}^{\\frac{2-y}{2}}dzdA = \\int\\int_D \\frac{2-y}{2}dA\\\\\n=\\int_0^1 \\int_{2x}^{2} \\frac{2-y}{2}dydx\\\\\n=x^2-2x+1\\\\\n=\\int _0^1\\left(x^2-2x+1\\right)dx\\\\\n=\\int _0^1x^2- \\int _0^12x+\\int _0^11dx\\\\\n=\\frac{1}{3}"


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