Calculate the line integral of the vector field along the line between the given points.

Question

Can someone give me some assistance on this problem? We are only solving line integrals geometrically.

$F = -5\vec{i} + 7\vec{j}$ from $(0,5)$ to $(3,16)$

At the moment, I am stuck with finding the limits of integration and $d\vec{r}$. Thanks

Answer 1

Since $\vec{F}(x,y)=\langle -5,7\rangle = \nabla f=\langle f_x, f_y \rangle$, where $f(x,y)=-5x+7y$, we use the Fundamental Theorem for line integrals to obtain: $$ \begin{align*} \int_C \vec{F} \cdot d\vec{r} &= \int_C \nabla f \cdot d\vec{r} \\ &= f(3,16)-f(0,5) \\ &= -5(3)+ 7(16)-(-5(0)+7(5)) \\ &= 62. \\ \end{align*} $$

Answer 2

We have

$$\begin{align} \int_C\vec F\cdot d\vec r&=\int_{(0,5)}^{(3,16)}(-5\vec i+7\vec j)\cdot (\vec i\,dx+\vec j\,dy)\\\\ &=-\int_0^3 5\,dx+\int_5^{16}7\,dy\\\\ &=62 \end{align}$$