Why does the primitive/anti derivative help define the area under a curve? (1 Viewer)

[mengfei]

Hey Bosers.
I've been reading ahead on Integration for next year's coursework in 2U, and so haven't had my teacher explain yet but it'd be awesome if someone could explain it in human speak haha!

When using the definite integral, we sub the X=a, X=b into the primitive and then subtract b from a, I Understand that.

But why are we subbing into the primitive?

Isn't the area under the curve defined by the curve X height, so why don't we just sub into that curve to give us the boundaries of the region?

I just don't understand why we use primitives for the calculation of the area under a curve if the curve is defined by curve height x width from X a to b in definite integral calculus

Thank you

 

[leehuan]

What does the height of a curve have to do with the area of a curve? Even for a rectangle the area is the height multipled by a length.
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If one proves the fundamental theorem of calculus then the relationship between the primitive and the area becomes more defined, because the definite integral IS the area under the curve.

 

[InteGrand]

Hey Bosers.
I've been reading ahead on Integration for next year's coursework in 2U, and so haven't had my teacher explain yet but it'd be awesome if someone could explain it in human speak haha!

When using the definite integral, we sub the X=a, X=b into the primitive and then subtract b from a, I Understand that.

But why are we subbing into the primitive?

Isn't the area under the curve defined by the curve X height, so why don't we just sub into that curve to give us the boundaries of the region?

I just don't understand why we use primitives for the calculation of the area under a curve if the curve is defined by curve height x width from X a to b in definite integral calculus

Thank you

You're right that in order to calculate the area, we'd have to do "height times width". But we'd need to take infinitesimally small widths in order to get the exact area. As your book should show, we do this by taking rectangles to approximate the area and then take the limit as the no. of rectangles goes to infinity. If this limit exists, the area is this limit.

The problem in practice is that these limits are usually hard to calculate. So in practice to calculate areas and integrals, we use the primitive, as this provides a practical or easier means of calculation than having to calculate a limit. The reason we can do this is as leehuan said due to a theorem called the Fundamental Theorem of Calculus, which basically says that if we integrate a 'nice' (continuous on [a, b]) function f(x) from a to b, the value of the integral is given by F(b) – F(a).

So the answer to question about why we use the primitive is that the Fundamental Theorem of Calculus says we can!

 

[InteGrand]

https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Physical_intuition