Consider the right tetrahedron cut from the first octant (i.e. x 0, y 0, and >...

 

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Consider the right tetrahedron cut from the first octant (i.e. x 0, y 0, and > 0 and the plane 6x+4y+2z = 12. 1. Solve for z in the equation of the plane to obtain the "top" formula for the integral. Because the solid is cut on the bottom by z = 0 (since it's cut from the first octant), the "bottom" formula will be z = 0. 2. Sketch the projection ("shadow") of the solid onto the xy-plane. This should be a 2d graph. Number the axes appropriately and shade in the region. 3. Use the graph and your answer from #1 to express the volume of the solid as a triple integral. 4. Evaluate your integral from #2 to find the volume of the tetrahedron. Consider the right tetrahedron cut from the first octant (i.e. x 0, y 0, and > 0 and the plane 6x+4y+2z = 12. 1. Solve for z in the equation of the plane to obtain the "top" formula for the integral. Because the solid is cut on the bottom by z = 0 (since it's cut from the first octant), the "bottom" formula will be z = 0. 2. Sketch the projection ("shadow") of the solid onto the xy-plane. This should be a 2d graph. Number the axes appropriately and shade in the region. 3. Use the graph and your answer from #1 to express the volume of the solid as a triple integral. 4. Evaluate your integral from #2 to find the volume of the tetrahedron.