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GATE 2016 EE - SET 2 - Question 9

The three variables x, y and z have to be shifted to represent a single parameter t.

Equation of line joining (0,0,0) and (1,1,1) is,

$\frac{x-0}{1-0}=\frac{y-0}{1-0}=\frac{z-0}{1-0}=t$

$x=y=z=t$ and $dx=dy=dz=dt$

Replacing the given equation using the parameter t and dt, we get,

$\int_{0}^{1}2t^{3}dt+\int_{0}^{1}2t^{3}dt+\int_{0}^{1}dt = \int_{0}^{1}(4t^{3}+1)$

$\Rightarrow [\frac{4t}{4}+t]_{0}^{1}\Rightarrow 1+1\Rightarrow 2$

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