Finding the volume of the tetrahedron using triple integralsD is the tetrahedron bounded by the coordinate planes and the plane 3 x + 3 y + z = 3, then express the volume of D as a triple integral.My Try:The z-limits are 0 ≤ z ≤ 3 − 3 x − 3 yIf y is the limit on the xy- plane then z = 0 0 ≤ y ≤ 1 − x.Similarly, 0 ≤ x ≤ 3.Finally, the integral will be ∫ 0 3 ∫ 0 1 − x ∫ 0 3 − 3 x − 3 y d z d y d x

Question

Finding the volume of the tetrahedron using triple integralsD  is the tetrahedron bounded by the coordinate planes and the plane   3  x  +  3  y  +  z  =  3, then express the volume of D as a triple integral.My Try:The z-limits are   0  ≤  z  ≤  3  −  3  x  −  3  yIf y is the limit on the xy- plane then   z  =  0   0  ≤  y  ≤  1  −  x.Similarly,   0  ≤  x  ≤  3.Finally, the integral will be       ∫          0              3            ∫          0              1      −      x            ∫          0              3      −      3      x      −      3      y        d  z     d  y     d  x

fairymischiefv9 · Accepted Answer

Explanation:It is correct for y and z bounds but for x we have-   y  =  z  =  0    ⟹    x  =  1    ⟹    0  ≤  x  ≤  1and therefore   V  =      ∫          0              1        d  x      ∫          0              1      −      x        d  y      ∫          0              3      −      3      x      −      3      y        d  z

Finding the volume of the tetrahedron using triple integrals. D is the tetrahedron bounded by the coordinate planes and the plane 3x+3y+z=3, then express the volume of D as a triple integral.

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