Find the volume of the given solid. The tetrahedron enclosed by the coordinate planes and the plane&nbsp;2x+y+z=4

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Find the volume of the given solid. The tetrahedron enclosed by the coordinate planes and the plane&amp;nbsp;2x+y+z=4

Clelioo · Accepted Answer

We need use a triple integral to seek out the amount of the given solid.The region is a tetraphon. Find the points p, q and r where the plane&amp;nbsp;2x+y+z=4&amp;nbsp;intercepts each of the x, y abd z axes, and use those points to sketch the region.&amp;nbsp;y=0,z=0:2x+(0)+(0)=4→p=(2,0,0)&amp;nbsp;x=0,z=0:2(0)+(0)+z=4→q=(0,4,0)&amp;nbsp;x=0,y=0:2(0)+(0)+z=4→r=(0,0,4)&amp;nbsp;Determine the outer bounds of integration in the x/y plane by setting z=0. The region is a triangle bounded by x=0, y=0, and y=4-2x&amp;nbsp;2x+y+(0)=4→y=4−2x&amp;nbsp;∫02∫04−2x&amp;nbsp;dy&amp;nbsp;&amp;nbsp;dx&amp;nbsp;&amp;nbsp;Now determine the inner bounds of integration by solving for z as a function of x and y&amp;nbsp;z=4−2x−y&amp;nbsp;∫02∫04−2x∫04−2x−y&amp;nbsp;dz&amp;nbsp;&amp;nbsp;dy&amp;nbsp;&amp;nbsp;dx&amp;nbsp;&amp;nbsp;Integrate with respect to z.&amp;nbsp;∫02∫04−2xz04−2x&amp;nbsp;dy&amp;nbsp;&amp;nbsp;dx&amp;nbsp;&amp;nbsp;∫02∫04−2x(4−2x−y)&amp;nbsp;dy&amp;nbsp;&amp;nbsp;dx&amp;nbsp;&amp;nbsp;Integrate with respect to z&amp;nbsp;∫02(4y−2xy−12y2)04−2x&amp;nbsp;dx&amp;nbsp;&amp;nbsp;∫024(4−2x)−2x(4−2x)−12(4−2x)2&amp;nbsp;dx&amp;nbsp;&amp;nbsp;∫02(16−8x)−(8x−4x2)−(8−8x+2x2)&amp;nbsp;dx&amp;nbsp;&amp;nbsp;∫022x2−8x+8&amp;nbsp;dx&amp;nbsp;&amp;nbsp;Integrate with respect to x.&amp;nbsp;&amp;nbsp;

Use a triple integral to find the volume of the given solid.The tetrahedron encl

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