Integration and The Fundamental Theorem of Calculus Is there an alternate way to answer this question, if anyone could please help.

The textbook answer it like this:


For given $x>0$ consider the triangle $T:=\{(t,u)\ |\ 0\leq u\leq t\leq x\}$ in the $(t,u)$-plane and on $T$ the function $\phi(t,u):= f(u)$; see the following figure.

By Fubini's theorem we can compute the integral
$$J:=\int_T \phi(t,u)\ {\rm d}(t,u)$$
in two ways, namely as
$$J=\int_0^x\left(\int_u^x \phi(t,u)\ dt\right)\ du=\int_0^x\left(\int_u^x f(u)\ dt\right)\ du=\int_0^x f(u)(x-u)\  du$$
and as
$$J=\int_0^x\left(\int_0^t \phi(t,u)\ du\right)\ dt=\int_0^x\left(\int_0^t f(u)\ du\right)\ dt\ .$$
