Question #224860

Evaluate the line integral of →{v}=xyi+y^{2}zj+e^{2z}k over the curve C whose parametric representation is given by x=t^{2},y=2t,z=t,0<t<1.

Expert's answer

Given that

"{v}=xyi+y^{2}zj+e^{2z}k\\\\\n x=t^{2},y=2t,z=t,0<t<1.\\\\"

Now the line integral is

"\\int_C Vdr\\\\\n{v(t)}=xyi+y^{2}zj+e^{2z}k\\\\\n\\space \\space \\space \\space \\space \\space \\space = 2t^3i+4t^3j +e^{2t}k\\\\\ndr = (2t,2,1)\\\\\n{v(t)}dr=2t(2t^3)+2(4t^3)+1(e^{2t})\\\\\n\\space \\space \\space \\space \\space \\space \\space \\space \\space \\space \\space \\space = 4t^4+8t^3+e^{2t}\\\\\n\\therefore \\int_C V.dr = \\int_0^1(4t^4+8t^3+e^{2t})dt\\\\\n(\\frac{4}{5}t^5+\\frac{8}{4}t^4+\\frac{1}{2}e^{2t})_0^1\\\\\n\\frac{23}{10}+\\frac{1}{2}e^2"

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