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## Main Question or Discussion Point

Hello,

I am currently attempting to cover rotational motion using Halliday's Fundamentals of Physics.

I understand very well the concept of moment of inertia as defined as the sum Σm

However, the textbook argues that if there are too many particles, the moment of inertia can be defined as an integral. And that is not something that I clearly understand.

I believe that my problem stems from a lack of understanding of the concept of integrals. I am currently taking a calculus class but the teacher hasn't yet covered this concept (although I believe I have a basic understanding of what it means thanks to some internet research).

In essence, what I don't understand is the following statement: an infinite sum can be expressed as an integral.

Could someone explain to me why this is true?

I faced the same problem when I covered the concept of center of mass (which can also be approached as an infinite sum when there are too many particles). I figured out, however, that if there were a function f(x

However, I have failed to approach the sum related to the moment of inertia in the same way. Indeed, f(m)=r

Could you help me figuring out why integrals can be used to express infinite sums? Was my approach to understand to the sum related to the center of mass incorrect?

Thank you very much,

(I don't know if I was right in posting my question in this section since it is more of a calculus question than a physics one; but since it involves physics concepts, I figured out it was legitimate to post it here. My apologies if I was wrong)

I am currently attempting to cover rotational motion using Halliday's Fundamentals of Physics.

I understand very well the concept of moment of inertia as defined as the sum Σm

_{i}*r_{i}^{2}.However, the textbook argues that if there are too many particles, the moment of inertia can be defined as an integral. And that is not something that I clearly understand.

I believe that my problem stems from a lack of understanding of the concept of integrals. I am currently taking a calculus class but the teacher hasn't yet covered this concept (although I believe I have a basic understanding of what it means thanks to some internet research).

In essence, what I don't understand is the following statement: an infinite sum can be expressed as an integral.

Could someone explain to me why this is true?

I faced the same problem when I covered the concept of center of mass (which can also be approached as an infinite sum when there are too many particles). I figured out, however, that if there were a function f(x

_{i})=m_{i}(giving the mass m_{i}of the_{i}th particle, the sum x_{i}m_{i}could indeed be expressed as an integral (since it would be the area under the curve).However, I have failed to approach the sum related to the moment of inertia in the same way. Indeed, f(m)=r

^{2}and f(r^{2}) are not actual functions since there could be several y's for a single x.Could you help me figuring out why integrals can be used to express infinite sums? Was my approach to understand to the sum related to the center of mass incorrect?

Thank you very much,

(I don't know if I was right in posting my question in this section since it is more of a calculus question than a physics one; but since it involves physics concepts, I figured out it was legitimate to post it here. My apologies if I was wrong)