6.2 Line integrals  (Page 7/20)

Note that curve C is the union of its four sides, and each side is smooth. Therefore C is piecewise smooth. Let $C_1$ represent the side from $\left(0,0\right)$ to $\left(2,0\right),$ let $C_2$ represent the side from $\left(2,0\right)$ to $\left(2,1\right),$ let $C_3$ represent the side from $\left(2,1\right)$ to $\left(0,1\right),$ and let $C_4$ represent the side from $\left(0,1\right)$ to $\left(0,0\right)$ ( [link] ). Then,

We want to compute each of the four integrals on the right-hand side using [link] . Before doing this, we need a parameterization of each side of the rectangle. Here are four parameterizations (note that they traverse C counterclockwise):

Therefore,

Notice that the value of this integral is positive, which should not be surprising. As we move along curve C ₁ from left to right, our movement flows in the general direction of the vector field itself. At any point along C ₁ , the tangent vector to the curve and the corresponding vector in the field form an angle that is less than 90°. Therefore, the tangent vector and the force vector have a positive dot product all along C ₁ , and the line integral will have positive value.

The calculations for the three other line integrals are done similarly:

and

Thus, we have ${\displaystyle {\int }_C\text{F}}·d\text{r}=2.$

To calculate the mass of the spring, we must find the value of the scalar line integral ${\displaystyle {\int }_C\left(e^x+yz\right)ds},$ where C is the given helix. To calculate this integral, we write it in terms of t using [link] :

Therefore, the mass is $\sqrt{5}\left(e^{\pi \text{/}2}+1\right)$ kg.