Answer to Question #314036 in Calculus for Jyo

Question #314036

Evaluate ∫c 𝐹. π‘‘π‘Ÿ,where 𝐹 = 𝑋^2 βˆ’ π‘Œ^2𝑖 + π‘₯𝑦𝑗 and curve 𝐢 is the arc of the curve 𝑦 = 𝑋^3 from (0,0) to (2,8).


1
Expert's answer
2022-03-21T02:04:29-0400

∫CFdr=[y=x3dy=3x2dxx:0β†’2F1=x2βˆ’y2=x2βˆ’x6F2=xy=x4]=∫02(x2βˆ’x6)dx+x4β‹…3x2dx==(x33+2x77)∣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}


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