Question

1)Sketch the vector field F⃗(x,y)=5i−4j and calculate the line integral of F⃗ along the line segment from (−5,−1) to (−3,0).
help anyone i dont know how to do this

Answer 1

the vector field is just an "arrow" in the direction 5 over, 4 down at every point.

the line integral is
\[ \int_C \vec{F}\cdot d\vec{r} \]
where \(\vec{r}\) is the position vector that describes the curve.

in this case r⃗ is a line given by the equation y = ½ x + 3/2
in other words,
\[ (\vec{r}= <x, ½ x + 3/2> \]
find the derivative of r⃗ with respect to x, and calculate
\[ \int \vec{F}\cdot \frac{d\vec{r}}{dx} \ dx \]

Answer 2

you should get
\[ \vec{F}= <5,-4> \\\frac{d\vec{r}}{dx} = <1,\frac{1}{2}> \]
and the integral
\[ \int_{-5}^{-3} <5,-4> \cdot < 1, \frac{1}{2}>\ dx \\
\int_{-5}^{-3} 3 \ dx = 3 x \bigg|_{-5}^{-3} = -9 +15 = 6
\]