How do I integrate a solid like where I am given 2 equations and using those 2 equations I am supposed to reflect them off the x or y axis and then create a solid and I am supposed to take a slice of that solid and write out the correct integration equation and then solve it. (Solving the integration part is not hard but getting to the correct equation is tough for me)
#Integrating solids
near lynx
fallen capeBOT
How do I integrate a solid like where I am given 2 equations and using those 2 e...
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vestal shale
How do I integrate a solid like where I am given 2 equations and using those 2 e...
show us an example
near lynx
show us an example
This is only one equation given but at times it is 2 equations
This would be an example for 2 equations
vestal shale
This would be an example for 2 equations
try it for a cylinder first
y = f(x) = some constant
near lynx
How does doing it via disc work
vestal shale
How does doing it via disc work
you cut it up into small discs horizontally and integrate over those
do you need to rigorously prove this works or just wanna know the technique and main idea
you add up the volume of these discs basically
at x, the radius is f(x) and let the thickness be dx then it's $\int_a^b\pi f(x)^2\diff x$
harsh heraldBOT
Coffey
near lynx
Pi r^2 is tbe equation for the area of a circle and we just plug f(x) - g(x) for the radius? Would g(x) be 0 because it is straight or would it need to be the height of y like in the picture it looks like it's a little below the y axis or a little over it.
vestal shale
wdym
you need the radius at every distance x on the x axis
which is? r(x) = |f(x) - g(x)|
sorry
this won't work because you have to get the area of the outer disk and subtract the area of the inner disc from it
really sorry for the mishap
near lynx
Ohhh yes that's correct
vestal shale
so A(x) = pi|f(x)^2-g(x)^2|dx
now integrate this
you could also think of it as doing it for f then g and subtracting the volume for g from that of f
near lynx
I understand
There is a difference between that problem and a problem like this one
How would I find the integral for this one? I understand I need to do something with h and essentially relate it to the radius
vestal shale
There is a difference between that problem and a problem like this one
this feels more like physics tho
how much work is being done to push water out the top?
what's the change in potential energy of the water
vestal shale
how much does the potential energy of the system change when a thin layer of water from the top (starting from depth x=0 but eventually it increases) of thickness dx perchance is pumped out?
it has a mass of : M(x) = pi * (radius at depth x)^2 * dx
and it travels x meters upward so the potential energy increases by xM(x)
so the work done is xM(x) considering we do it infinitesimally slowly so as to not develop kinetic energy which isn't relevant to the question probably
can you do the rest?
near lynx
This is complete different language, this is supposed to be physical applications in calculus, it's a work problem but we use integrals and don't really account potential or kinetic energy, I was just seeing if someone in this server knew how to solve a problem like this because essentially you're supposed to use the variables to your advantage
vestal shale
This is complete different language, this is supposed to be physical application...
well I can't bring myself to be blind about the physics when it says "calculate the work done"
you could think of it as "force" pushing out that "disc"
but that is the change in potential energy to
the force due to gravity downward is M(x)g
we push it against gravity x distance anti parallelly
so the work done is M(x)g times x here, it's positive
near lynx
Okay thank you
vestal shale
Okay thank you
if you are done then do +close and read what it says (no one reads it please read it...)
hazy pagodaBOT
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