- #1

Scott77

- 7

- 0

## Homework Statement

I need to find the work done by the force field:

$$\vec{F}=(5x-8y\sqrt{x^2+y^2})\vec{i}+(4x+10y\sqrt{x^2+y^2})\vec{j}+z\vec{k}$$

moving a particle from a to b along a path given by:

$$\vec{r}=\frac{1}{2}\cos(t)\vec{i}+\frac{1}{2}\sin(t)\vec{j}+4\arctan(t)\vec{k}$$

## The Attempt at a Solution

So I set up my line integral:

$$\vec{F}(\vec{r}(t))=(\frac{5}{2}\cos(t)-2\sqrt{2}\sin(t))\vec{i}+(2\cos(t)+\frac{5\sqrt{2}}{2}\sin(t))\vec{j}+(4\arctan(t))\vec{k}$$

$$\vec{r'}(t)=\left(-\frac{1}{2}\sin(t)\right)\vec{i}+\left(\frac{1}{2}\cos(t)\right)\vec{j}+\left(\frac{4}{t^2+1}\right)\vec{k}$$

$$\int_0^{1}\left(-\frac{5+5\sqrt{2}}{4}\sin(t)\cos(t)+\sqrt{2}\sin^2(t)+\cos^2(t)+\frac{16\arctan(t)}{t^2+1}\right)\; \text{d}t=5.86436$$

I have left a lot of steps out, it gets messy! Could this problem be solved by reducing the integral to polar coordinates?