How do we convert a summation to integration?

[MacNCheese]

When converting a summation of the form

$$\sum x_i y_i$$

to integration, how do we know if it's

$$\int x dy$$

or

$$\int y dx$$

At first I thought they're equivalent but obviously that's only true for a linear function with no constant offset.

I kind of see integration as a better form of multiplication so I've always had trouble with the fact that multiplication is commutative but integration isn't. A little help?

 

[mathman]

Your description of the summation as an analog of integration is not quite accurate.
∑x_iΔy_i -> ∫xdy
∑y_iΔx_i -> ∫ydx

 

[jhooper3581]

Summation is used to find the area under the curve. Integration is same thing, but much faster.

 

[MacNCheese]

Well if you look at something like center of mass, it's mentioned as (in wikipedia)

$$\frac {\sum m_i r_i} {\sum m_i}$$

for discrete particles and

$$\frac {1} {M} \int r dm$$

for continuous distribution. There's no 'delta' anywhere.

 

[HallsofIvy]

Yes, those are two completely different formulas. The first is NOT the Riemann sum leading to the second.

 

[mathman]

Well if you look at something like center of mass, it's mentioned as (in wikipedia)

$$\frac {\sum m_i r_i} {\sum m_i}$$

for discrete particles and

$$\frac {1} {M} \int r dm$$

for continuous distribution. There's no 'delta' anywhere.

Although the symbol Δ doesn't appear, it is implicit in the meaning of m_i which is a small piece of the mass, which sums to M.

 

[MacNCheese]

Yes, those are two completely different formulas. The first is NOT the Riemann sum leading to the second.

Isn't it? Then how do we derive such an expression?

Although the symbol Δ doesn't appear, it is implicit in the meaning of m_i which is a small piece of the mass, which sums to M.

Thanks, that helps.