Confused how to use calculus in physics

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The discussion clarifies the use of integrals in physics, particularly in calculating electric fields from charge distributions. It emphasizes that the integral ∫dE cosθ represents the sum of differential contributions to the electric field, where dE is a vector and cosθ projects it onto the x-axis. The need for integration arises from having an infinite number of charge points, contrasting with simpler cases like point charges where superposition applies. The goal is to express the variables appropriately, either in terms of E or θ, based on the geometry of the situation. Overall, understanding the integral's role is crucial for accurately summing contributions in physics problems.
jaydnul
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I understand simple concepts, like \frac{dx}{dt}=v and why that is, but when I'm doing, for example, uniform charge distributions, I don't understand what the integral is actually doing. For example:

E_x=∫dEcosθ

From what I learned in calculus, the dE means with respect to. So when taking an integral you usually have the form ∫y(x)dx and the interval is [a,b], which are x values.

Why isn't the integral above in that form then? I mean at the very least, ∫dθcosθ would make more sense to me.
 
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Why isn't it in that form? Because you haven't made it into that form yet, that is your goal. You need to express E in terms of theta, or theta in terms of E, by looking at the geometry of the situation.

Every little point in a charge distribution contributes to the overall electric field. If you just had 2 point charges you would add the fields in accordance with superposition. But now that you have an infinite number of points in a larger distribution, you need to do an integral to add them all up.
 
Jd0g33,

∫ means a "sum" over differential amounts.

In ∫dE cosθ the differential amount is dE cosθ

dE is a vector and dE x cosθ is its projection on the x-axis.

Adding up all the projections of every dE, you get E_{x}.

When taking ∫y(x)dx, the differential amount being added up is y(x)dx, that is, y(x) times dx.
This is the "area" under the point y(x).

In this explanation, I have used some loose terms, but I hope I could pass the message.
 
So I know that electrons are fundamental, there's no 'material' that makes them up, it's like talking about a colour itself rather than a car or a flower. Now protons and neutrons and quarks and whatever other stuff is there fundamentally, I want someone to kind of teach me these, I have a lot of questions that books might not give the answer in the way I understand. Thanks
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