# finding work by integrals

• Mar 23rd 2009, 10:57 AM
s7b
finding work by integrals
Find the work done by the force F from (0,0,0) to (1,1,1) over the following path:

The path of the line segment from (0,0,0) to (1,1,0) followed by the line segment from (1,1,0) to (1,1,1) where the Force F=(1/(x^2+1))j

How do I set up this integral??
• Mar 23rd 2009, 12:33 PM
Work done by variable force
Hello s7b
Quote:

Originally Posted by s7b
Find the work done by the force F from (0,0,0) to (1,1,1) over the following path:

The path of the line segment from (0,0,0) to (1,1,0) followed by the line segment from (1,1,0) to (1,1,1) where the Force F=(1/(x^2+1))j

How do I set up this integral??
The first line segment is in the plane $\displaystyle z=0$; the force is in the direction of the $\displaystyle y$-axis, and it moves along the line $\displaystyle y = x$ in the plane from $\displaystyle (0,0)$ to $\displaystyle (1,1)$. So as the point of application of the force moves from $\displaystyle (x,y)$ to $\displaystyle (x+\delta x,y+\delta y), \delta y = \delta x \Rightarrow$ work done = $\displaystyle F\delta y = F\delta x$.
To find the work done, then, you'll need to evaluate $\displaystyle \int_0^1F\,dx$, where $\displaystyle F = \frac{1}{x^2+1}$
The second part of the movement is parallel to the $\displaystyle z$-axis, and is therefore at right angles to the force. So it does a zero amount of work.