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INTEGRATION [Solved!]

My question

Must the differential be included ALWAYS in the integrand?

If so, then why?

Relevant page

1. Integration: The General Power Formula

What I've done so far

Take, for instance, Example 1.

If dx was not written after the integrand, is that acceptable?

We know it is a derivative and that we are integrating because of the integral sign that is why we figured it is okay to not write it.

X

Must the differential be included ALWAYS in the integrand?

If so, then why?
Relevant page

<a href="/methods-integration/1-integration-power-formula.php">1. Integration: The General Power Formula</a>

What I've done so far

Take, for instance, Example 1.

If dx was not written after the integrand, is that acceptable?

We know it is a derivative and that we are integrating because of the integral sign that is why we figured it is okay to not write it.

Re: INTEGRATION

Yes, the differential must always be included!

Firstly, it tells us the variable we are integrating by. For example, what if I have more than one (potential) variable in the integration but with no differential, something like:

`int p^2q`

This could mean:

`int p^2qdp = p^3/3q+K` (where the `q` is a constant and `p` is the variable)

or

`int p^2qdq = p^2q^2/2 + K` (where `p` is now a constant, and `q` is the variable)

This would get very messy (and you'd get things wrong all the time) when you get up to double (and triple) integrals, like the ones you'll see in this page: Double Integrals

Secondly, the differential is an essential part of the concept of integration. The idea of finding an (exact) area under the curve is to break it up into rectangles, `f(x)` high and `Deltax` wide. When we let those rectangle widths get smaller and smaller (to infinitely thin) and we add them, we get the exact area. This is what's happening in Area Under a Curve page. The `dx` is the way we indicate we have been adding those rectangles which were `Deltax` wide.

X

Yes, the differential must always be included!

Firstly, it tells us the variable we are integrating by. For example, what if I have more than one (potential) variable in the integration but with no differential, something like:

`int p^2q`

This could mean:

`int p^2qdp = p^3/3q+K` (where the `q` is a constant and `p` is the variable)

or

`int p^2qdq = p^2q^2/2 + K` (where `p` is now a constant, and `q` is the variable)

This would get very messy (and you'd get things wrong all the time) when you get up to double (and triple) integrals, like the ones you'll see in this page: <a href="http://tutorial.math.lamar.edu/Classes/CalcIII/DIGeneralRegion.aspx">Double Integrals</a>

Secondly, the differential is an essential part of the concept of integration. The idea of finding an (exact) area under the curve is to break it up into rectangles, `f(x)` high and `Deltax` wide. When we let those rectangle widths get smaller and smaller (to infinitely thin) and we add them, we get the exact area. This is what's happening in <a href="https://www.intmath.com/integration/3-area-under-curve.php">Area Under a Curve</a> page. The `dx` is the way we indicate we have been adding those rectangles which were `Deltax` wide.

Re: INTEGRATION

Thank you for an extremely concise response! Awesome!

X

Thank you for an extremely concise response! Awesome!

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