blackstreet23
  • blackstreet23
What was the calculus formula to determine the volume of a solid? I don't mean solid of revolution. Just regular solid.
Mathematics
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
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blackstreet23
  • blackstreet23
volumes of cross sections
freckles
  • 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
  • blackstreet23
yes that is it ! thank you !

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freckles
  • freckles
that's also the formula used in solids formed by revolution of some curve about some axis
LynFran
  • LynFran
u forget to square it..i think @freckles
freckles
  • freckles
you know where we take circles as the cross section
freckles
  • freckles
no A(x) is the area you don't want to square the area
LynFran
  • LynFran
o ok
freckles
  • freckles
if the cross section is a circle then A(x)=pi*r^2
freckles
  • freckles
where r is a function of x
freckles
  • freckles
though we could look at things in terms of y too not just x
freckles
  • freckles
you know depending on what suits us better
blackstreet23
  • 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
  • LynFran
i think the f(x) is actually the radius
blackstreet23
  • blackstreet23
f(x) is the function
blackstreet23
  • blackstreet23
i mean the height of the funtion
LynFran
  • LynFran
yes which is intern the radius so A=pi*r^2
LynFran
  • LynFran
and freckles just use the Area instead...i think
blackstreet23
  • blackstreet23
I guess i am kind of confuse lol
LynFran
  • LynFran
ok lets wait on till @freckles come online...
freckles
  • freckles
what's the question?
freckles
  • freckles
area of a circle is what I called A(x)
freckles
  • freckles
or the are of the cross section is A(x)
freckles
  • 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
  • freckles
|dw:1440894543729:dw| Say we want to take this curve on [a,b] and rotate it about the x-axis
freckles
  • 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
  • freckles
|dw:1440894662165:dw| so we have a cross section that is a circle
freckles
  • freckles
notice the distance from the x-axis to the curve is indeed the radius which is called f(x) in this case
freckles
  • freckles
\[\int\limits_{a}^{b}A(x) dx \\ \text{ so we have } A(x)=\pi \cdot (f(x))^2 \\ \]
LynFran
  • LynFran
so i was right about f(x) being the radius?
blackstreet23
  • blackstreet23
ohh i see
freckles
  • freckles
|dw:1440894830121:dw| yes
freckles
  • freckles
but the A(x) is not squared
freckles
  • freckles
A(x) just means area of the cross section
blackstreet23
  • blackstreet23
ohh i see ! I thought f(x) was just the height of the function
blackstreet23
  • blackstreet23
It makes so much sense now :)
freckles
  • freckles
well you can also say |f(x)| is the height of the function
freckles
  • freckles
I put | | around it because f(x) could happen to be negative which just means the height is below the x-axis
LynFran
  • LynFran
cool..

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