Evaluate β«c πΉ. ππ,where πΉ = π^2 β π^2π + π₯π¦π and curve πΆ is the arc of the curve π¦ = π^3 from (0,0) to (2,8).
β«CFdr=[y=x3dy=3x2dxx:0β2F1=x2βy2=x2βx6F2=xy=x4]=β«02(x2βx6)dx+x4β 3x2dx==(x33+2x77)β£02=82421\int_C{Fdr}=\left[ \begin{array}{c} y=x^3\\ dy=3x^2dx\\ x:0\rightarrow 2\\ F_1=x^2-y^2=x^2-x^6\\ F_2=xy=x^4\\\end{array} \right] =\int_0^2{\left( x^2-x^6 \right) dx+x^4\cdot 3x^2dx}=\\=\left( \frac{x^3}{3}+\frac{2x^7}{7} \right) |_{0}^{2}=\frac{824}{21}β«CβFdr=β£β‘βy=x3dy=3x2dxx:0β2F1β=x2βy2=x2βx6F2β=xy=x4ββ¦β€β=β«02β(x2βx6)dx+x4β 3x2dx==(3x3β+72x7β)β£02β=21824β
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