Answer to Question #223887 in Chemical Engineering for Lokika

Question

Evaluate the line integral of ∀ =xyl+ y^{2}zj+e^{2z}k over the curve C whoseparametric representation is given by x=t^{2},y=2t,z=t,0<t<1

Answer 1

$\{V\}=xyi+ y^{2}zj+e^{2z}k\\ x=t^{2},y=2t,z=t,0<t<1$

Now the line integral is $\int_CVdr$

$\{V(t)\}=t^2(2t)i+ (2t)^{2}tj+e^{2t}k\\ =2t^3i+ 4t^{3}j+e^{2t}k\\ dr= (2t,2,1)\\ \{V(t)\}*dr=(2t^3i+ 4t^{3}j+e^{2t}k)(2t,2,1)=(4t^4+8t^3+e^{2t})\\ \int_CVdr= \int_0^1(4t^4+8t^3+e^{2t})dt\\ \mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx\\ =\int _0^14t^4dt+\int _0^18t^3dt+\int _0^1e^{2t}dt\\ =\frac{14}{5}+\frac{e^2-1}{2}$