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