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, 

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

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