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I've encountered many integrals that seem to integrate functions of distance with respect to mass, for example, $\int_0^Mr^2dm$ for the moment of inertia of continuous mass distribution.

I'm not sure what an integral of a function of distance with respect to mass means mathematically, but I have a couple of ideas.

$r$ is a function of mass: Then the integral can be expressed as $\int_0^Mr(m)^2dm$, which is definitely possible to integrate using calculus, however, I'm not sure what this would mean intuitively.

In calculus, I was taught the single integral of a function could be thought of as the "area between the graph and the x-axis". More rigorously, this would be the supremum of the set of all possible lower Darboux sums, equal to the infinium of all possible upper Darboux sums, which is equal to the limit of a Reimann sum as the number of intervals approaches infinity. All of these definitions require distance from the axis at some mass $m_0$ to be defined, however, I don't see how we'd define say $m=3$ or $r(3)$. Wouldn't there be multiple ways to define the location of the point where $m=m_0$ for any $m_0$ (I'm assuming the distance from the axis is determined based on the position of that mass)?

$dm$ is shorthand for another expression: This is the second thought I had, maybe dm is shorthand for something like $\rho dr$ where $\rho$ is the density of the object. This idea came from the fact that in math, $ds$ is often used for shorthand for $\sqrt{(u'(t)+v'(t)}dt$ or something similar for higher dimensions. If this is the case, the idea expressed by the integral makes much more sense, since integrating with respect to distance is much more intuitive than integrating with respect to mass.

But this idea still has a couple of issues. The limits of integration are defined in terms of mass, implying that the integral is actually with respect to mass- not distance (I've always seen shorthand like $ds$ is used within integrals that had limits in terms of the parameter, not s). Also, the idea of multiplying infinitesimals like algebraic variables seems somewhat non-rigorous, that is, it's not obvious what saying something like $dm=\rho dx$ means mathematically. I'm aware solving integrals by using substitution or solving differential equations by "multiplying both sides by dx"$dx$" is possible; however, these are both notational conveniences for the chain rule and there doesn't seem to be a similar mechanism at play here.

So what does integrating with respect to mass actually mean? Is it a combination of both of these ideas? Is it neither?

Any clarification would be greatly appreciated.

I've encountered many integrals that seem to integrate functions of distance with respect to mass, for example, $\int_0^Mr^2dm$ for the moment of inertia of continuous mass distribution.

I'm not sure what an integral of a function of distance with respect to mass means mathematically, but I have a couple of ideas.

$r$ is a function of mass: Then the integral can be expressed as $\int_0^Mr(m)^2dm$, which is definitely possible to integrate using calculus, however, I'm not sure what this would mean intuitively.

In calculus, I was taught the single integral of a function could be thought of as the "area between the graph and the x-axis". More rigorously, this would be the supremum of the set of all possible lower Darboux sums, equal to the infinium of all possible upper Darboux sums, which is equal to the limit of a Reimann sum as the number of intervals approaches infinity. All of these definitions require distance from the axis at some mass $m_0$ to be defined, however, I don't see how we'd define say $m=3$ or $r(3)$. Wouldn't there be multiple ways to define the location of the point where $m=m_0$ for any $m_0$ (I'm assuming the distance from the axis is determined based on the position of that mass)?

$dm$ is shorthand for another expression: This is the second thought I had, maybe dm is shorthand for something like $\rho dr$ where $\rho$ is the density of the object. This idea came from the fact that in math, $ds$ is often used for shorthand for $\sqrt{(u'(t)+v'(t)}dt$ or something similar for higher dimensions. If this is the case, the idea expressed by the integral makes much more sense, since integrating with respect to distance is much more intuitive than integrating with respect to mass.

But this idea still has a couple of issues. The limits of integration are defined in terms of mass, implying that the integral is actually with respect to mass- not distance (I've always seen shorthand like $ds$ is used within integrals that had limits in terms of the parameter, not s). Also, the idea of multiplying infinitesimals like algebraic variables seems somewhat non-rigorous, that is, it's not obvious what saying something like $dm=\rho dx$ means mathematically. I'm aware solving integrals by using substitution or solving differential equations by "multiplying both sides by dx" is possible; however, these are both notational conveniences for the chain rule and there doesn't seem to be a similar mechanism at play here.

So what does integrating with respect to mass actually mean? Is it a combination of both of these ideas? Is it neither?

Any clarification would be greatly appreciated.

I've encountered many integrals that seem to integrate functions of distance with respect to mass, for example, $\int_0^Mr^2dm$ for the moment of inertia of continuous mass distribution.

I'm not sure what an integral of a function of distance with respect to mass means mathematically, but I have a couple of ideas.

$r$ is a function of mass: Then the integral can be expressed as $\int_0^Mr(m)^2dm$, which is definitely possible to integrate using calculus, however, I'm not sure what this would mean intuitively.

In calculus, I was taught the single integral of a function could be thought of as the "area between the graph and the x-axis". More rigorously, this would be the supremum of the set of all possible lower Darboux sums, equal to the infinium of all possible upper Darboux sums, which is equal to the limit of a Reimann sum as the number of intervals approaches infinity. All of these definitions require distance from the axis at some mass $m_0$ to be defined, however, I don't see how we'd define say $m=3$ or $r(3)$. Wouldn't there be multiple ways to define the location of the point where $m=m_0$ for any $m_0$ (I'm assuming the distance from the axis is determined based on the position of that mass)?

$dm$ is shorthand for another expression: This is the second thought I had, maybe dm is shorthand for something like $\rho dr$ where $\rho$ is the density of the object. This idea came from the fact that in math, $ds$ is often used for shorthand for $\sqrt{(u'(t)+v'(t)}dt$ or something similar for higher dimensions. If this is the case, the idea expressed by the integral makes much more sense, since integrating with respect to distance is much more intuitive than integrating with respect to mass.

But this idea still has a couple of issues. The limits of integration are defined in terms of mass, implying that the integral is actually with respect to mass- not distance (I've always seen shorthand like $ds$ is used within integrals that had limits in terms of the parameter, not s). Also, the idea of multiplying infinitesimals like algebraic variables seems somewhat non-rigorous, that is, it's not obvious what saying something like $dm=\rho dx$ means mathematically. I'm aware solving integrals by using substitution or solving differential equations by "multiplying both sides by $dx$" is possible; however, these are both notational conveniences for the chain rule and there doesn't seem to be a similar mechanism at play here.

So what does integrating with respect to mass actually mean? Is it a combination of both of these ideas? Is it neither?

Any clarification would be greatly appreciated.

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What Does It Mean To Integrate With Respect To Massdoes it mean to integrate with respect to mass?

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What Does It Mean To Integrate With Respect To Mass?

I've encountered many integrals that seem to integrate functions of distance with respect to mass, for example, $\int_0^Mr^2dm$ for the moment of inertia of continuous mass distribution.

I'm not sure what an integral of a function of distance with respect to mass means mathematically, but I have a couple of ideas.

$r$ is a function of mass: Then the integral can be expressed as $\int_0^Mr(m)^2dm$, which is definitely possible to integrate using calculus, however, I'm not sure what this would mean intuitively.

In calculus, I was taught the single integral of a function could be thought of as the "area between the graph and the x-axis". More rigorously, this would be the supremum of the set of all possible lower Darboux sums, equal to the infinium of all possible upper Darboux sums, which is equal to the limit of a Reimann sum as the number of intervals approaches infinity. All of these definitions require distance from the axis at some mass $m_0$ to be defined, however, I don't see how we'd define say $m=3$ or $r(3)$. Wouldn't there be multiple ways to define the location of the point where $m=m_0$ for any $m_0$ (I'm assuming the distance from the axis is determined based on the position of that mass)?

$dm$ is shorthand for another expression: This is the second thought I had, maybe dm is shorthand for something like $\rho dr$ where $\rho$ is the density of the object. This idea came from the fact that in math, $ds$ is often used for shorthand for $\sqrt{(u'(t)+v'(t)}dt$ or something similar for higher dimensions. If this is the case, the idea expressed by the integral makes much more sense, since integrating with respect to distance is much more intuitive than integrating with respect to mass.

But this idea still has a couple of issues. The limits of integration are defined in terms of mass, implying that the integral is actually with respect to mass- not distance (I've always seen shorthand like $ds$ is used within integrals that had limits in terms of the parameter, not s). Also, the idea of multiplying infinitesimals like algebraic variables seems somewhat non-rigorous, that is, it's not obvious what saying something like $dm=\rho dx$ means mathematically. I'm aware solving integrals by using substitution or solving differential equations by "multiplying both sides by dx" is possible; however, these are both notational conveniences for the chain rule and there doesn't seem to be a similar mechanism at play here.

So what does integrating with respect to mass actually mean? Is it a combination of both of these ideas? Is it neither?

Any clarification would be greatly appreciated.