Area between two curves as a definite integral

Partition $[a,b]$ into $n$ subintervals $[x_{i-1},x_i]$ of width $\Delta x=(b-a)/n$, and pick a sample point $x_i^*$ in each. The strip of the region over $[x_{i-1},x_i]$ has height approximately $|f(x_i^*)-g(x_i^*)|$, so its area is approximately $|f(x_i^*)-g(x_i^*)|\,\Delta x$.

Summing over all $n$ strips gives the Riemann sum $\sum_{i=1}^n |f(x_i^*)-g(x_i^*)|\,\Delta x$, which approximates the true area of $R$ by construction.

As $n\to\infty$, the error in each strip's approximation goes to $0$ by uniform continuity of $f-g$ on the compact interval $[a,b]$. Since $|f-g|$ is continuous, the Riemann sum converges to $\int_a^b|f(x)-g(x)|\,dx$ by the definition of the Riemann integral. Hence $S(R)=\int_a^b|f(x)-g(x)|\,dx$.

When $f(x)\ge g(x)$ throughout $[a,b]$, this simplifies to $S(R)=\int_a^b[f(x)-g(x)]\,dx$ with no absolute value needed; when the sign of $f-g$ changes at points inside $(a,b)$, splitting the integral at those points and choosing the correct order of subtraction on each piece gives the practical computation method.