How do I evaluate the line integral $$\int _c \mathbf{F}\cdot\mathrm{d}\mathbf{r}$$ where $\mathbf{F} = x^2\mathbf{i} + 2y^2\mathbf{j}$ and $c$ is the curve given by $\mathbf{r}(t)=t^2\mathbf{i} + t\mathbf{j}$ for $t \in [0,1]$.
I have started with: $\int \mathbf{F}(t^2\mathbf{i} + t\mathbf{j})\cdot \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\,\mathrm{d}t$ but im not sure if this is right.
In order to work out the integral, you need to find first the derivatives of the coordinates of the curve. Set $\mathbf{r} = x\mathbf{i} + y \mathbf{j} = (x,y)$. Then $x(t) = t^2$ and $y(t) = t$, thus $ \dot{x} = 2t$ and $\dot{y} = 1$. Then $$ \begin{split} \int_c \mathbf{F} \cdot \mathrm{d}\mathbf{r} &= \int_0^1 \left(x\times\frac{\mathrm{d}x}{\mathrm{d}t} + y\times\frac{\mathrm{d}y}{\mathrm{d}t}\right)\mathrm{d}t \cr &= \int_0^1 (t^2 \times 2t + 2t \times 1)\mathrm{d}t \cr &= \int_0^1 (2t^3 + 2t)\mathrm{d}t \cr &= \frac{1}{2} + 2\times\frac{1}{2} = \frac{3}{2} \end{split} $$