Differentiate circle area/sphere volume with respect to its radius variable to get circles circumference /spheres surface area.
In analogy,
For cubes and Squares , in their integrals ,the infinitesimal change (da) for standard variable "side length(a)" has to be substituted by it's equivalent expression interms of "half of side length(s) ".
ds- is actually in the direction of gradient of the level surfaces/curves of cubes/squares.
i.e One should consider, square area as 4s^2 where square's side (a)=2*s
Cube's volume as 8s^3 where cube's side (a)=2*s
Now differentiating the above formulae for perimeter/surface area , one gets 8s & 24s^2 as answers.
As last step, convert them back into 'a' to check with standard results by doing s--->a/2
Surface area of cube=24(a/2)^2=6a^2
perimeter of square =8(a/2) =4a
In analogy,
For cubes and Squares , in their integrals ,the infinitesimal change (da) for standard variable "side length(a)" has to be substituted by it's equivalent expression interms of "half of side length(s) ".
ds- is actually in the direction of gradient of the level surfaces/curves of cubes/squares.
i.e One should consider, square area as 4s^2 where square's side (a)=2*s
Cube's volume as 8s^3 where cube's side (a)=2*s
Now differentiating the above formulae for perimeter/surface area , one gets 8s & 24s^2 as answers.
As last step, convert them back into 'a' to check with standard results by doing s--->a/2
Surface area of cube=24(a/2)^2=6a^2
perimeter of square =8(a/2) =4a
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