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## Main Question or Discussion Point

Hello, I am a medical doctor but interested in mathematics and physics. Consider a parabola with its equation y = x

dy/dx = 2x

and dy = 2x.dx

If we make small right angled triangles under the parabola with the base dx then the height of the triangles will be dy. The hypotenuse (dh) of the triangles will be

dh

dh

dh

if we integrate dh it will give us the sum of squares of the small pieces of the curve, which is parabola here (S

S

But I have been told that it is not allowed in calculus to integrate dx squared. Can some one please shed some light on it? Thank you.

^{2}dy/dx = 2x

and dy = 2x.dx

If we make small right angled triangles under the parabola with the base dx then the height of the triangles will be dy. The hypotenuse (dh) of the triangles will be

dh

^{2}= (dx)^{2}+ (dy)^{2}dh

^{2}= (dx)^{2}+ (2x.dx)^{2}dh

^{2}= (dx)^{2}(1 + 4x^{2})if we integrate dh it will give us the sum of squares of the small pieces of the curve, which is parabola here (S

_{h}^{2}).S

_{h}^{2}= ∫ 1 + 4x^{2}.dx^{2}But I have been told that it is not allowed in calculus to integrate dx squared. Can some one please shed some light on it? Thank you.

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