- #1

- 68

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Thus, why is the area of a rectangle or square- lxb or sxs???

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- Thread starter physio
- Start date

- #1

- 68

- 1

Thus, why is the area of a rectangle or square- lxb or sxs???

- #2

- 806

- 23

what if the strips are made smaller and smaller such that the strips are infinitesimally small then the formula doesn't make any sense

Obviously the formula will not make sense if you compute it completely differently.

Of course, it's entirely possible to add together all of those infinitesimally small strips, which will give you the right answer; you just have to use calculus.

The approach you described (using strips of elements unit 1 wide) seems perfectly intuitive to me and clearly describes why the area of the square is computed as it is, so I'm not sure what else you want. You could use calculus and integrate over a square to the find the area, I suppose, which would give you the formula s

We want to compute the area of a square with side length

[tex]\int_0^a\! a \, \mathrm{d} x \: = \: a \cdot a - a \cdot 0\: = \: a^2[/tex]

What we've done here is exactly what you described; we took each of those slices and shrunk them down until they were infinitesimally small, and then added them up. This gives us the right answer.

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- #3

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Hm.

If width of single strip is identically zero, then there is nothing that could be done. However, if strip width is identically zero, then width is not infinitesimal. So Your reasoning got astray at the moment You passed from "infinitesimal" to "identically zero". You might have not noticed this transition. If infinitesimal, then not zero. Yes, as close to zero as You like. Never zero though. Use limits. Or integral, of course. I guess question was of speculative nature.

This reminds me of Zeno paradox.

Cheers.

- #4

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- #5

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I don't know about Spivak's Calculus. I do know about a ton of other books I've read on calculus, though. There are 2 types of calculus books. First type are books for students of mathematics and theoretical physics. They have to know it rigorously because it's in the curriculum and profs lecture it that way. Second type of calculus books are books written for engineers. Those written for engineers go straight for the head: they aim at calculating things. Engineer books are not concerned with purely theoretical aspects of calculus. So, one might go for second type books first, and when differential details become clear, one is advised to take a byte at the real thing. Otherwise, if not informed on the subject at all, one is easily lost in all the details of theoretical math. And calculus is not easy one way or another. It's a huge area and one never gets to master it entirely. Ever. Finally, in my opinion, reading only one book on calculus is not enough. The beast is too huge for only one weapon.

Cheers.

- #6

- 68

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Thanks...! I will look into these things.

- #7

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- #8

- 2,967

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Then, to prove the formula for the area of a rectangle:

http://optimizingke.com/wp-content/uploads/2010/05/Inner-Area-to-Outer-Area.bmp

The side of the outer square is [itex]a + b[/itex], and the side of the inner square is [itex]a - b[/itex] (assuming [itex]a > b[/itex]). Then, the area of the outer square is the sum of the areas of the inner square and the four identical rectangles:

[tex]

(a + b)^2 = (a - b)^2 + 4 A

[/tex]

[tex]

A = \frac{(a + b)^2 - (a - b)^2}{2}

[/tex]

[tex]

A = \frac{(a^2 + 2 a b + b^2) - (a^2 - 2 a b + b^2)}{4}

[/tex]

[tex]

A = a \, b

[/tex]

- #9

- 68

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Thanks for your replies...!!!!

- #10

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If so, you can think of it as taking the 2cm along one side and multiplying it by how many of those there are. one side is 2cm wide, and stretches over 2cm. If you are asking how you get 4cm

substitute cm into "d" you then have [itex]{(2d)^2}={2^2}{d^2}=4{d^2}[/itex]

plug our dimension back into d's place, you get:

[tex]{2cm{\cdot}2cm}=4{cm^2}[/tex]

Another example is acceleration. In algebriac terms acceleration is:

[itex]\frac{{v_f}-{v_i}}{t}=a[/itex] where:

v

v

t=time

a=acceleration.

there are fundemental units of quantity; here is the wiki page for the SI Units http://en.wikipedia.org/wiki/International_System_of_Units#Units_and_prefixes

We take these fundemental units to make others, such as these: http://en.wikipedia.org/wiki/List_of_physical_quantities

notice the velocity is [itex]\frac{d}{t}[/itex], time is (fundemntal)=t, and acceleration is [itex]\frac{d}{t^2}[/itex].

You may ask, "why is the time squared for acceleration?" treat the dimensions algebriaclly.

[tex]\frac{(\frac{d}{t})}{t}=\frac{d}{tt}=\frac{d}{t^2}[/tex]

The quantities' dimensions treat each other algebriaclly.

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