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The Fundamental Theorem of Calculus is of course correct, and you are applying it correctly. The statements the rate of change of work with respect to displacement is force and the instantaneous rate of change of work with respect to displacement is force are correct. The thing to keep in mind is that it's not work that is instantaneous, but its ...


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No matter how ridiculous you may find it, it is true. The most general definition of work is indeed that the infinitesimal work done along an infinitesimal path is just force times the length of the path, i.e. $$ \mathrm{d}W = F\mathrm{d}s$$ Therefore, the amount of work done along a path in space $\gamma : [a,b] \to \mathbb{R}^3$ is the line integral $$ ...


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Where are you getting these books? Here it makes absolutely no sense to mention indefinite integrals and I have never seen such confusing statements. You need definite integral, usually in multiple dimensions (ideally 3D, as we live in 3D space, but you can simplify in some cases). The integration goes over the entire volume of the object, you have ...


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I think the book authors might have the Huygens-Steiner theorem in mind when saying that. The theorem says that if the body has moment of inertia $I_{cm}$ with respect to axis crossing its center of mass, then moving the reference axis to another parallel axis gives a new moment of inertia $I'$, related to original one by $$I'=I_{cm}+md^2,$$ where $d$ is ...


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Unless I have the rest of the book, it is difficult to understand what he meant. But many books have equations that could make not much sense or whose explanations are very unclear (no author is perfect). The only thing you really need to know is what you already seem to know. In an actual calculation you use a definite integral, and the constant $C$ doesn't ...


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As it looks like another question I've supplied an answer to might be duplicated here (and hence closed), I am going to provide a similar but not identical answer here. In words - divergence is the flux of something into or out of a closed volume, per unit volume. The best visual picture I have of this is a fluid flow. Imagine water spewing out of a tap - ...


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The integral $\int$ is indeed a continuous version of summation. There are two ways of looking at this: As an indefinite integral, you have $\int df(x)=\int F(x)dx$. As we area dealing with indefinite integral, the right side after evaluation still depends on $x$. So naturally, it's a function of $x$. On the left, the notation with $(x)$ is perhaps ...


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Let's say for simplicity $F(x) = 3 x^2$, just so we have an example to talk about. As you wrote $$f(x) = \int F(x) \mathrm d x + C = x^3 +C\,.$$ And now the initial conditions come into play (the $x_0$, $v_0$ and so on) or in some cases boundary conditions. Those are always needed to find a specific solution to a differential equation. So if I also know, ...


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The job of calculus is to handle quantities that vary over the domain of the problem at hand. Often, and particularly in introductory physics, we care about quantities that vary in time. We cannot put them into our equations as constants. We also often care about the interrelationship of these quantities. So for example now velocity is given by ...


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How else can you recover displacement from a non-linear velocity without integrating? How else would you calculate acceleration of a non-linear velocity without differentiating? ADDED: If your question is how to solve $\displaystyle v(t) = \frac{dx}{dt} = b\sqrt{x}$, let us know.



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