How can you integrate a vector without decomposing it in x and y components?

[mad]

Hello, I just wanted to know if you can integrate a vector, and if so, how.
Here is a problem: A charged piece of metal of linear density 2 * 10^-6 * x , between x=2 and x=5.
I found its charge by integrating.. and it is 21 micro C.

Now that's what I want to know: calculate the electric field at x=0
I know that $$\vec{E} = \int{} d \vec{E}$$

but in my physics book, it says I have to decompose it in $$E_x$$ or $${E_y}$$ and integrate. But since I hate working without vectors, do you think there's a way to integrate without decomposing it in x and y, like something like this..:
$$\vec{E} = \int{} \frac{k dq }{r^2} \vec{u}$$ = $$\vec{E} = \int{} \frac{k \lambda dx }{x^2} \vec{u}$$

Is there a way to do this? My teacher hasn't started this topic yet, I'm just curious.
Thanks a lot !

 

[dextercioby]

Hello, I just wanted to know if you can integrate a vector, and if so, how.
Here is a problem: A charged piece of metal of linear density 2 * 10^-6 * x , between x=2 and x=5.
I found its charge by integrating.. and it is 21 micro C.

Now that's what I want to know: calculate the electric field at x=0
I know that $$\vec{E} = \int{} d \vec{E}$$

but in my physics book, it says I have to decompose it in $$E_x$$ or $${E_y}$$ and integrate. But since I hate working without vectors, do you think there's a way to integrate without decomposing it in x and y, like something like this..:
$$\vec{E} = \int{} \frac{k dq \vec{u}}{r^2}$$ = $$\vec{E} = \int{} \frac{k \lambda dx \vec{u}}{x^2}$$
Is there a way to do this? My teacher hasn't started this topic yet, I'm just curious.
Thanks a lot !

What is the direction of the unit vector $\vec{u}$ ?

Daniel.

 

[mad]

What is the direction of the unit vector $\vec{u}$ ?

Daniel.

It is -i , since they ask the electric field at x=0 . Is there a reason you are asking this?

 

[dextercioby]

Yes,i wanted to know whether it was constant or not.If it is,then u can take it outta the integral and compute the "scalar" integral (which yields the component of $\vec{E}$ along the direction of $\vec{u}$)...

Daniel.

 

[mad]

Yes,i wanted to know whether it was constant or not.If it is,then u can take it outta the integral and compute the "scalar" integral (which yields the component of $\vec{E}$ along the direction of $\vec{u}$)...

Daniel.

$$k \lambda \vec{i} \int \frac{dx}{x^2}$$

You mean this? We have just started integrals in math., so I may have confused some things.