(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Ok so I am going to go a bit outside of the normal procedure to post a question (again) because I don't understand the example nor the first problem fully. I will post the example first and any questions I have with what is being done will be posted as the example progresses, ok? ok

So the example states:

Find the volume of a pyramid of height h whose base has area B, as in Figure 6.1.7 which can be found http://images2e.snapfish.com/232323232fp537;4>nu=52::>379>256>WSNRCG=32<:7;9279347nu0mrj"

2. Relevant equations

Here is the entire theorem as given in the book:

Let B(u,w) be a real function of two variables that has the addition property in the interval [a,b] i.e.,

[tex] B(u,w) = B(u,v) + B(v,w) \;\; for \;\; u<v<w \;\; in \;\; [a,b] [/tex]

Suppose h(x) is a real function continuous on [a,b] and for any infinitesimal subinterval:

[tex] [x, x +\Delta x] \; of \; [a,b], [/tex]

[tex] \Delta B = h(x) \Delta x [/tex]

Then B(a,b) is equal to the integral:

[tex] B(a,b) = \int^{b}_{a} h(x) dx [/tex]

(I suppose right here I should state that I have not had any experience with a function of two variables. I have done this book chapter by chapter and this was never introduced, so I am kind of flying blind here already.)

3. The attempt at a solution

Ok so the example goes on to say:

Place the pyramid on its side with the apex at x=0 and the base at x=h.

(First problem, they seem to have switched where the apex and base are because from the picture you can clearly see that the base is set to 0)

We use the fact that at any point x between 0 and h, the cross section has area proportional to x^2, so that:

(Second problem, how do we know that [0,h] cross sections have area proportional to x^2? this is not mentioned in early chapters or prior to this example in any way)

[tex] \frac {A(x)}{x^{2}} = \frac {B}{h^{2}} [/tex]

[tex] A(x) = \frac {Bx^{2}}{h^{2}} [/tex]

(Third problem, why did we set this up this way? Did I forget something crucial from geometry?)

Then the volume is:

[tex] V = \int^{h}_{0} \frac {Bx^{2}}{h^{2}} dx = \frac {1}{3} * \frac {Bh^{3}}{h^{2}} = \frac {1}{3} Bh [/tex]

(Thankfully I understand what they did here completely, its just getting to this point...)

_____________________________________________________________________________

So the first problem of the chapter asks:

The base of a solid is the triangle in the x,y plane with vertices at (0,0), (0,1), and (1,0). The cross sections perpendicular to the x-axis are squares with one side on the base. Find the volume of the solid.

For some reason I really can't fathom how to draw this, I will post this now and attempt to draw a picture of what they are talking about. So if you reply before I post some kind of picture for the problem above just focus on my example questions.

Thanks for any and all help and understanding.

EDIT: http://images2e.snapfish.com/232323232fp53838>nu=52::>379>256>WSNRCG=32<:7;<;58347nu0mrj" is a picture of what I think they are saying in problem one, am I close?

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# Homework Help: Infinite Sum Theorem - Help Understanding

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