# Can't use a simple integral properly-must be retarded

• cantRemember
If you cut out a small length at position x, and measure the mass: it is Δm, and the length is Δx. The average linear density is Δm/Δx. If you cut out shorter and shorter pieces that ratio tends to the density at point x, to dm/dx = f(x). The total mass is the integral of f(x).f

#### cantRemember

I will show you my (obviously wrong) way of thinking when i have to apply an integral.
Please correct me where I'm wrong.

(imaginary question)
Suppose you have a mass distribution across a line, where the mass of each point is given by the equation f(x)=a*x (assume a is a constant)
find the total mass, if the line is c meters long (beggining @ x=0)

(My stupid train of thought)
We have to add up all the individual masses on the line.
so divide the line into n segments, calculate the mass for each one and add them up
f(x1)+f(x2)+...+f(xn)=
a*x1+a*x2+...+a*xn=
a(x1+x2+...+xn)=
a*c

What am i doing wrong?
(Besides the language use, I'm not a native and have little experience in such teminology)

I will show you my (obviously wrong) way of thinking when i have to apply an integral.
Please correct me where I'm wrong.

(imaginary question)
Suppose you have a mass distribution across a line, where the mass of each point is given by the equation f(x)=a*x (assume a is a constant)
find the total mass, if the line is c meters long (beggining @ x=0)

(My stupid train of thought)
We have to add up all the individual masses on the line.
so divide the line into n segments, calculate the mass for each one and add them up
f(x1)+f(x2)+...+f(xn)=
a*x1+a*x2+...+a*xn=
a(x1+x2+...+xn)=
a*c

What am i doing wrong?
(Besides the language use, I'm not a native and have little experience in such teminology)

Mass at each point does not have much sense. Is not f(x) the density at point x? Then the mass dm of a line element around x is dm=f(x) dx and you have to integrate from 0 to c.

If f(x) means the mass from 0 to x, then the total mass at x=c is just f(c) ehild

Mass at each point does not have much sense. Is not f(x) the density at point x? Then the mass dm of a line element around x is dm=f(x) dx and you have to integrate from 0 to c.

If f(x) means the mass from 0 to x, then the total mass at x=c is just f(c) ehild

Thank you. But how can you define density in segments while you cannot define mass?

a(x1+x2+...+xn)=
a*c
......
(x1+x2+...+xn)≠c

x1=0
xcenter=c/2
xn=c

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Thank you. But how can you define density in segments while you cannot define mass?

If you have a continuous line the mass of a point of it is zero.
But you can cut out a small length at position x, and measure the mass: it is Δm, and the length is Δx. The average linear density is Δm/Δx. If you cut out shorter and shorter pieces that ratio tends to the density at point x, to dm/dx = f(x). The total mass is the integral of f(x).

ehild

Last edited:
The linear density at point x, f(x), is the mass per unit length. Think of it as a chain.