Find $$\int_{C}\mathbf{F}\,{\boldsymbol\cdot}\, d\mathbf{r}$$

if $\mathbf{F}(x,y)=x\mathbf{i}+xy\mathbf{j}$ and the curve $C$ is traced out by the vector function $\mathbf{r}(t)=t\,\mathbf{i}\,+\,t^{2} \mathbf{j}$, $0\leq t\leq 2$.

Solution Parametric equations of the curve $C$ are $$x(t) =t\qquad y(t) =t^{2}\qquad 0\leq t\leq 2$$

So, $$\mathbf{F}=x\mathbf{i}+xy\,\mathbf{j}=t\mathbf{i}+(t) (t^{2}) \mathbf{j}=t\mathbf{i}+t^{3}\mathbf{j}$$

and $$d\mathbf{r}=\dfrac{d\mathbf{r}}{dt}\,dt=\dfrac{d}{dt}( t\,\mathbf{i} +t^{2}\mathbf{j})\, dt=(\mathbf{i}+2t\mathbf{j})\,dt$$

Then $$\mathbf{F}\,{\boldsymbol\cdot}\, d\mathbf{r}=( t\mathbf{i}+t^{3}\,\mathbf{j}) \,{\boldsymbol\cdot}\, ( \mathbf{i}+2t\,\mathbf{j}) \,dt=( t+2t^{4}) \,dt$$

so that $$\int_{C}\mathbf{F}\,{\boldsymbol\cdot}\, d\mathbf{r}=\int_{0}^{2}(t+2t^{4}) \,dt=\dfrac{74}{5}$$