Calc1.9

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Let f\, be a continuous function for x\geq a\,.
Show that \int _{a}^{x}\int _{a}^{s}f(y)\,dy\,ds=\int _{a}^{x}f(y)(x-y)\,dy\,

Let F(s)=\int _{a}^{s}f(y)\,dy\,

\int _{a}^{x}\int _{a}^{s}f(y)\,dy\,ds=\int _{a}^{x}F(s)\,ds\,

=\int _{a}^{x}{\frac  {d}{ds}}sF(s)\,ds\,

=xF(x)-aF(a)-\int _{a}^{x}sF'(s)\,ds\,

=x\int _{a}^{x}f(y)\,dy-\int _{a}^{x}yf(y)\,dy\,

=\int _{a}^{x}f(y)(x-y)\,dy\,

Calculus

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