Question

) Calculate integral R C F · dr · dr, where F(x, y, z) =< 2x ln y, x 2 y + z 2 , 2yz >, and C is a curve with a parameterization r(t) =< t2 , t, t >, 1 ≤ t ≤ e

Answer 1

$$r'(t)=\lt2t,1,1\gt$$

$$F(r(t))=\lt2t^2\ln t, t^3+t^2,2t^2\gt$$

$$F(r(t))r'(t)=4t^3\ln t+t^3+t^2+2t^2$$

$$\int_CF\cdot dr=\displaystyle\int_{1}^{e}(4t^3\ln t+t^3+3t^2)dt$$

$$\int4t^3\ln t dt$$

$$u=\ln t, du=\dfrac{1}{t}dt$$

$$dv=4t^3dt, v=t^4$$

$$\int4t^3\ln t dt=t^4\ln t-\int t^3dt=t^4\ln t-\dfrac{t^4}{4}+C$$

$$\int_CF\cdot dr=\displaystyle\int_{1}^{e}(4t^3\ln t+t^3+3t^2)dt$$

$$=[t^4\ln t-\dfrac{t^4}{4}+\dfrac{t^4}{4}+t^3]\begin{matrix} e \\ 1 \end{matrix}$$

$$=e^4+e^3-1$$