What was the calculus formula to determine the volume of a solid? I don't mean solid of revolution. Just regular solid.

- blackstreet23

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- blackstreet23

volumes of cross sections

- freckles

Are you sure you aren't looking for:
\[\text{ Volume }=\int\limits_a^bA(x) dx \text{ where } A(x) \text{ is the area of the cross section } ?\]

- blackstreet23

yes that is it ! thank you !

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- freckles

that's also the formula used in solids formed by revolution of some curve about some axis

- LynFran

u forget to square it..i think @freckles

- freckles

you know where we take circles as the cross section

- freckles

no A(x) is the area
you don't want to square the area

- LynFran

o ok

- freckles

if the cross section is a circle then A(x)=pi*r^2

- freckles

where r is a function of x

- freckles

though we could look at things in terms of y too
not just x

- freckles

you know depending on what suits us better

- blackstreet23

but how is that the formula for solids of revolution? I mean the formula for solids of revolution is \[\int\limits_{a}^{b}(\pi*f(x)^2) dx\]

- LynFran

i think the f(x) is actually the radius

- blackstreet23

f(x) is the function

- blackstreet23

i mean the height of the funtion

- LynFran

yes which is intern the radius
so A=pi*r^2

- LynFran

and freckles just use the Area instead...i think

- blackstreet23

I guess i am kind of confuse lol

- LynFran

ok lets wait on till @freckles come online...

- freckles

what's the question?

- freckles

area of a circle is what I called A(x)

- freckles

or the are of the cross section is A(x)

- freckles

if the cross section is a circle then A(x) is pi*r^2
where r is the radius
f(x) is the distance from the curve to the x-axis
so f(x) is the radius if the axis of rotation is the x-axis

- freckles

|dw:1440894543729:dw|
Say we want to take this curve on [a,b] and rotate it about the x-axis

- freckles

first reflecting the curve about the x-axis gives us: |dw:1440894612771:dw|
now that drawing is not that great but that curve is basically suppose to be a mirror for the other curve I drew

- freckles

|dw:1440894662165:dw|
so we have a cross section that is a circle

- freckles

notice the distance from the x-axis to the curve is indeed the radius which is called f(x) in this case

- freckles

\[\int\limits_{a}^{b}A(x) dx \\ \text{ so we have } A(x)=\pi \cdot (f(x))^2 \\ \]

- LynFran

so i was right about f(x) being the radius?

- blackstreet23

ohh i see

- freckles

|dw:1440894830121:dw|
yes

- freckles

but the A(x) is not squared

- freckles

A(x) just means area of the cross section

- blackstreet23

ohh i see ! I thought f(x) was just the height of the function

- blackstreet23

It makes so much sense now :)

- freckles

well you can also say |f(x)| is the height of the function

- freckles

I put | | around it because f(x) could happen to be negative
which just means the height is below the x-axis

- LynFran

cool..

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