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Answer to Question #223887 in Chemical Engineering for Lokika

Question #223887

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

1
2021-08-31T23:57:18-0400

"\\{V\\}=xyi+ y^{2}zj+e^{2z}k\\\\\nx=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\\\\\n=2t^3i+ 4t^{3}j+e^{2t}k\\\\\ndr= (2t,2,1)\\\\\n\\{V(t)\\}*dr=(2t^3i+ 4t^{3}j+e^{2t}k)(2t,2,1)=(4t^4+8t^3+e^{2t})\\\\\n\\int_CVdr= \\int_0^1(4t^4+8t^3+e^{2t})dt\\\\\n\\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\\\\\n=\\int _0^14t^4dt+\\int _0^18t^3dt+\\int _0^1e^{2t}dt\\\\\n=\\frac{14}{5}+\\frac{e^2-1}{2}"

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