Integral of gradient dotted with dr Let $f(x,y) = x^2 y$ and let $C$ be curve $\vec{r}(t) = \langle e^{t^2},\ \sin(\pi t/2 )\rangle$ for $0\le t \le 1$ 
$$\int_C \nabla f \cdot d\vec{r}$$

Well, we have that
$$\int_{C}\nabla f\cdot d\mathbf{r}=\int_{t_{i}}^{t_{f}}\nabla f(\mathbf{r}(t))\cdot\frac{d\mathbf{r}}{dt}dt=\int_{t_{i}}^{t_{f}}\frac{df(\mathbf{r}(t))}{dt}dt$$
or
$$\int_{C}\nabla f\cdot d\mathbf{r}=f(\mathbf{r}(t_{f}))-f(\mathbf{r}(t_{i}))$$
So, in this particular case $f(\mathbf{r})\equiv f(x,y)=x^2y$; $\mathbf{r}(t)\equiv (x(t),y(t))=(e^{t^2},\sin \frac{\pi t}{2})$; and $t_i=0$, $t_f=1$;
$$\int_{C}\nabla f\cdot d\mathbf{r}=f(\mathbf{r}(1))-f(\mathbf{r}(0))=f(e,1)-f(1,0)$$
or
$$\int_{C}\nabla f\cdot d\mathbf{r}=e^2\cdot1-1^2\cdot 0=e^2$$
