GATE 2016 EC (EC3) – Question 38
A triangle in the xy plane is bounded by $2x=3y$, y=0 and x=3. The volume above it and below $x+y+z=6$ is ________.
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Correct answer: 10.0 to 10.0
Explanation
The triangle has 0≤x≤3 and 0≤y≤2x/3. The height is z=6−x−y, which is positive over this triangle.
Integrate:
$$V=\int_0^3\int_0^{2x/3}(6-x-y)\,dy\,dx.$$
The inner result is $4x-8x^2/9$. Integrating gives $18-8=$ **10 cubic units**.