A number of solutions of length of curve functions I have run across are solved to a point and then instructed to "simplify the integrand before going further" as in the following example: f(y) = x3 - 1/4x3   on [1,2] interval integrate  [ ( 1 + (x3 -1/4x3) )2 ]1/2 dx diferentiate the function, and square it simplify the intergrand to [ 1 + x6 - 1/2 + 1/16x6 dx ]1/2 I am absolutly puzzled as to where the middle term, - 1/2, comes from! Would you please give me a step-wise understanding of this feature?

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.