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help with surface integral please?

#mathematics #calculus #integral #calculus3 #surfaceintegral

i just need a hint how to get this started.

Let T be the triangle formed by the plane 3x+3y+2z=6 in first octant. If mass is distributed on T with density p(x,y,z) = 2(x+y+z) (mass/unit area), find the total mass of the triangle.

Answers (1)

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r7kcmudbaa profile image
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This should just be a regular triple integral.

The bounds for z are clear: 0 ≤ z ≤ 3 - 3x/2 - 3y/2.

Projecting this into the xy-plane yields 3x + 3y = 6 in the first quadrant

==> 0 ≤ y ≤ 2 - x with 0 ≤ x ≤ 2.

So, m = ∫∫∫ p(x,y,z) dV.

= ∫(x = 0 to 2) ∫(y = 0 to 2-x) ∫(z = 0 to 3 - 3x/2 - 3y/2) 2(x + y + z) dz dy dx.

I hope this helps!

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