# Integration Intuition for work between x1 and x2

1. Feb 24, 2013

### Joshb60796

1. The problem statement, all variables and given/known data
A force F=bx3 acts in the x direction, where the value of b is 3.7N/m3. How much work is done by this force in moving an object from x=0.00m to x=2.6m?

2. Relevant equations
W=F*D
∫undu = (u(n+1)/(n+1))+C

3. The attempt at a solution
I know from previous problems that I'm going to integrate to find the area under the curve bx3 from 0 to 2.6 but I don't have the intuition as to why. I don't really understand why I'm doing this besides the face that I see a function raised to a power and not much more information is given. I'm following a pattern based on a pattern I see. I get that the force is changing in the y coordinate and that the x coordinate I'm worried about goes from 0 to 2.6 and I'm finding the area using integration but is the force a jerk because it's raised to the 3rd power, would it be acceleration if raised to the second power? I remember doing derivatives and thinking this. How did they come up with b? How did b come to be? What measuring device gives a reading like 3.7N/m3? I'm not even sure I'm asking the right questions but I want to fully understand this.

Last edited: Feb 24, 2013
2. Feb 24, 2013

### rock.freak667

If you consider a force F moving its point of application an infinitesimal distance 'dx', then the work done within in this distance is dW = F dx

For the work done over the entire distance from x2 to x1

$$\int_{0} ^W 1 dW = \int_{x_1} ^{x_2} F dx \Rightarrow W = \int_{x_1} ^{x_2} F dx$$

Graphically, this represents the area under the force-distance curve.

What could happen to get such a formula is that based on measuring force with distance, you could find that F∝x3 such that F= bx3. If you plot a graph of F against x3, the gradient would be 'b'.

To get what units 'b' would have, the units of the product of 'bx3' would need to be Newtons (N); x is in m.

so N = b*m3 such that b = N/m3. So you would not always have an instrument measuring something in N/m3 but you might be able to figure it out based on other data.

3. Feb 24, 2013

### Joshb60796

Thank you sir for your insight. What was that symbol you used after F? I've never seen it before. Also, what is a gradient exactly? Is it like a coefficient that changes? I've only had up to Calculus 2 so far.

4. Feb 24, 2013

### rock.freak667

The ∝ ? I am not sure why it appears so small but it is the symbol for 'directly proportional to' so as one increases, the other increases by a set amount.

Gradient refers to a rate of change. In the straight line equation y=mx+c, the gradient would be equal to m. Such that the change in the distance y (denoted by Δy) divided by the change in the distance x (Δx) is equal to m.

Δy/Δx = m