Question

Evaluate ∫c 𝐹. 𝑑𝑟,where 𝐹 = 𝑋^2 − 𝑌^2𝑖 + 𝑥𝑦𝑗 and curve 𝐶 is the arc of the curve 𝑦 = 𝑋^3 from (0,0) to (2,8).

Answer 1

$\int_C{Fdr}=\left[ \begin{array}{c} y=x^3\\ dy=3x^2dx\\ x:0\rightarrow 2\\ F_1=x^2-y^2=x^2-x^6\\ F_2=xy=x^4\\\end{array} \right] =\int_0^2{\left( x^2-x^6 \right) dx+x^4\cdot 3x^2dx}=\\=\left( \frac{x^3}{3}+\frac{2x^7}{7} \right) |_{0}^{2}=\frac{824}{21}$