Integral of a vector

Question

In many cases mostly in physics I have seen that while integrating vectors we take their specific components and we integrate does who add together in one direction and we can not do anything about the components due to a continuous body which are in different direction and are not getting simply added. So does this mean that an integral can simply add things ? Because if it were able to find the net we would have simply integrated the vector of each infinitesimal without taking components? Am i correct? In the case of a ring, only the cos component is integrated. If the integral also takes the direction in consideration then if we simply integrated the electric field and not take the components

Answer 1

The reason why you integrate only one component (in this case $\cos\theta dE$) is that in the other directions you can simply say that due to symmetry the integral will cancel.

The superposition principle says that you can decompose a vector in components, and the sum of the components is the vector. So if you want to add together all $d\vec E$ vectors you get $$\vec E=\int d\vec E=\hat x\int dE_x+\hat y\int dE_y+\hat z\int dE_z=\hat xE_x+\hat yE_y+\hat zE_z$$

In this problem, let's say that $z$ is along the horizontal axis. Then due to symmetry, the integral in the $x$ and $y$ direction will cancel. But that is not always the case. You can calculate the electric field from half of the ring. Then you will have additional components.