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Let $f:[a,0]->lR$ a continuous function.Show that

$∫$(0 to x)($∫$(0 to y) $f(t)dt)dy=∫$(0 to x)$[(x-y)f(t)]dt$, $x,y\in [0,a]$

My attempt

the region of integration from the double integral is t=0 to t=y with y\in [0,x]

This can be rewritten as y=t to y=x in [0,x]

So interchanging the order of integration yields

$∫$(0 to x)($∫$(0 to y) $f(t)dt)dy=$∫(0 to x)($∫$(t to x) f(t)dy)dt

=$∫$(0 to x) $[yf(t)]$ (for y=t to x) dt

=$∫$(0 to x) $[(x-t)f(t)]dt$

Anyone can help me find the error in my procedure please?

Thanks for your help

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1 Answer

up vote 1 down vote accepted

It may be a typo in the task formulation. LHS depends only on $x$ but RHS depends on $x$ and $t$. All of your steps are right: $$\int\limits_{0}^{x}\left(\int\limits_{0}^{y}{f(t)dt} \right)dy=\int\limits_{0}^{x}\left(\int\limits_{t}^{x}{f(t)dy} \right)dt=\int\limits_{0}^{x}\left(f(t) \int\limits_{t}^{x}{dy} \right)dt=\int\limits_{0}^{x}\left(f(t) (x-t) \right)dt$$

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