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## Homework Statement

Hi everybody! I would like to make sure I properly solved that problem because I find the result strange:

Given a force field ##F_x = axy^3##, ##F_y = bx^2y^2##, ##F_z = cz^3##.

Calculate the work with the line integral ##\int_{C} \vec{F} \cdot d\vec{r}## from point ##P_1(1,0,0)## to ##P_2(0,1,1)## along:

a) a line;

b) an helix by the z-axis.

## The Attempt at a Solution

Okay so first I parametered the line:

[tex]\vec{r}(\tau) = (1-\tau, \tau, \tau) \mbox{ with } \tau \in [0,1][/tex]

Is the first component of my vector correct? I have a little doubt about it.

[tex]\implies \dot{\vec{r}}(\tau) = (-1,1,1) \\

\implies A = \int_{0}^{1} \vec{F} \cdot \vec{r} \cdot d\tau \\

= \int_{0}^{1} (a (1 - \tau) \tau^3, b(1-\tau)^2\tau^2, c\tau^3)(-1,1,1)d\tau \\[/tex]

Here again, I have a little doubt about what I did with the force field vector. Is that correct?

[tex]\int_{0}^{1} (-a\tau^3 + a\tau^4 + b\tau^2 - 2b\tau^3 + b\tau^4 + c\tau^3) d\tau \\

a[\frac{\tau^5}{5} - \frac{\tau^4}{4}]_0^1 + b[\frac{\tau^5}{5} + \frac{\tau^3}{3} - \frac{\tau^4}{2}]_0^1 + c[\frac{\tau^4}{4}]_0^1 \\

A = -\frac{a}{20} + \frac{b}{30} + \frac{c}{4}[/tex]

So yeah here is my result for (a). I'd like to know if I use the right method before I attempt to solve (b).

What do you guys think?

Thanks a lot in advance for your answers and suggestions, I appreciate it.

Julien.

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