Question

* The value of the line integral
$\int (2x{y}^{2}dx+2x^{2}ydy+dz)$ along a path joining the origin (0,0,0) and the point (1, 1, 1)

• A. 0
• B. 2
• C. 4
• D. 6

Answer 1

The correct option is B 2

I= $\int (2x{y}^{2}dx+2x^{2}ydy+dz)$
Path joining (0,0,0) and (1,1,1) can be written as
$\frac{x-0}{1-0}=\frac{y-0}{1-0}=\frac{z-0}{1-0}=\lambda$
which is equation of straight line in 3D.
$\Rightarrow x=y=z=\lambda$
$\Rightarrow dx=dy=dz$ and whenx =y =z = 0 $\Rightarrow$ $\lambda$ = 0 and when x = y = z = 1 $\Rightarrow$ $\lambda$ = 1
I = ${\int }_0^1(2{\lambda }^3+2{\lambda }^3+1)d\lambda$
$={\int }_0^1(4{\lambda }^3+1)d\lambda ={[{\lambda }^4+\lambda ]}_0^1=2$