Question

Evaluate LaTeX: \int_cF.dr\:\: where LaTeX: F\left(x,y,z\right)=xzi-yzkF(x,y,z)=xzi−yzk and c is the line segment from (3,0,1) to (-1,2,0)

Answer 1

Given,

$$F(x,y,z)=xz \textbf i−yz \textbf k \ Nd \ line \ segment \ c\ from\ (3,0,1)\to(-1,2,0)$$

Let,

$$r(t) =(1-t)<3,0,1>+t<-1,2,0>=<3-4t,2t,1-t>$$

Differentiate with respect to t,

$$dr=(-4\textbf i+2\textbf j-\textbf k)dt$$

Now,

$$F(r(t))=(3-4t)(1-t)i-2t(1-t)k$$

$$F(r(t))=(3t^2-7t+3)\textbf i+(2t^2-2t)\textbf k$$

Therefore,

$$\int_cF.dr=[(3t^2-7t+3)\textbf i+(2t^2-2t)\textbf k]\cdot[-4\textbf i+2\textbf j-\textbf k]dt\\$$

$$=\int_{0}^{1}-4(3t^2-7t+3)-(2t^2-2t)dt\\ =\int_{0}^{1}-14t^2+30t-12dt\\ =[-\frac{14}{3}t^3+15t^2-12t]_{0}^{1}\\ =-\frac{5}{3}$$

Thus, the required answer is $\frac{5}{3}$ .