Call the parabola [tex]P[/tex], parameterized by [tex]\mathbf r(y)=\langle y^2+1,y\rangle[/tex] with [tex]0\le y\le 1\rangle[/tex]. Then the work done by [tex]\mathbf f(x,y)[/tex] along [tex]P[/tex] is
[tex]\displaystyle\int_P\mathbf f(x,y)\cdot\mathrm d\mathbf r=\int_{y=0}^{y=1}\mathbf f(x(y),y)\cdot\dfrac{\mathrm d\mathbf r(y)}{\mathrm dy}\,\mathrm dy[/tex]
[tex]=\displaystyle\int_0^1\langle(y^2+1)^2,ye^{y^2+1}\rangle\cdot\langle2y,1\rangle\,\mathrm dy[/tex]
[tex]=\displaystyle\int_0^1(2y^5+4y^3+2y+ye^{y^2+1})\,\mathrm dy=\frac73-\frac e2+\frac{e^2}2[/tex]
