Sketching the change in a cube's volume

  • Thread starter wisredz
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Hi all,
I've come by an interesting while studying. Here it goes

The volume [itex]V=x^3[/itex] of a cube of with edges of length x increases by an amount [itex]\Delta V[/itex] when x increases by an amount [itex]\Delta x [/itex]. Show with a sketch how to represent [itex]\Delta V[/itex] geometrically as the some of the volumes of

(a) Three slabs of dimensions x by x by [itex]\Delta x [/itex]
(b) Three bars of dimensions x by [itex]\Delta x [/itex] by [itex]\Delta x [/itex]
(c) One cube of dimensions [itex]\Delta x [/itex] by [itex]\Delta x [/itex] by [itex]\Delta x [/itex]

The differential formula [itex]dV=3x^2*dx[/itex] estimates the change in V with three slabs.

Well that is kinda interesting right? Why is it so? I think the rest (3 bars and a cube) is the error in the estimate. It it right?
 

HallsofIvy

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Yes, that's true and I agree that it is interesting!
 

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