Question

Find the work done by the force field f in moving an object from a to

Answer 1

Assuming the path from $a$ to $b$ is a line segment, we can parameterize it by
$\mathbf r(t)=(1-t)(1,1)+t(3,9)=(1+2t,1+8t)=(x(t),y(t))$
for $0\le t\le1$. Then the work done by $\mathbf f(x,y)$ along this path, which I'll denote $\mathcal C$, is
$\displaystyle\int_{\mathcal C}\mathbf f(x,y)\cdot\mathrm d\mathbf r=\int_{t=0}^{t=1}f(x(t),y(t))\cdot(2,8)\,\mathrm dt$$=\displaystyle\int_0^1(2(1+8t)^{3/2},3(1+2t)(1+8t))\cdot(2,8)\,\mathrm dt$$=\displaystyle4\int_0^1(1+8t)^{3/2}\,\mathrm dt+24\int_0^1(1+10t+16t^2)\,\mathrm dt$
$=\dfrac{242}5+272=\dfrac{1602}5$