Multivariable Calculus Question?
set up and evaluate an integral which computes the volume bounded by the planes x+y+z=3, z=0, y=0, x=0, y=1, x=2.
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set up and evaluate an integral which computes the volume bounded by the planes x+y+z=3, z=0, y=0, x=0, y=1, x=2.
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Verified answer
The volume ∫∫∫ 1 dV equals
∫(x = 0 to 2) ∫(y = 0 to 1) ∫(z = 0 to 3 - x - y) 1 dz dy dx
= ∫(x = 0 to 2) ∫(y = 0 to 1) (3 - x - y) dy dx
= ∫(x = 0 to 2) [(3 - x)y - y^2/2] {for y = 0 to 1} dx
= ∫(x = 0 to 2) (5/2 - x) dx
= (5x/2 - x^2/2) {for x = 0 to 2}
= 3.
I hope this helps!