Knowlpedia

Additivity of the Riemann integral over subintervals

Splitting a Riemann integral at an interior point

Let f:[a,b]→Rf:[a,b]\to\mathbb R be and let c∈[a,b]c\in[a,b]. The restrictions of ff to [a,c][a,c] and [c,b][c,b] are Riemann integrable, and

∫abf(x) dx=∫acf(x) dx+∫cbf(x) dx.\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx.

For c=ac=a or c=bc=b, interpret the integral over the degenerate interval as zero.

Example

Splitting at c=1c=1 gives

∫02x dx=∫01x dx+∫12x dx=12+32=2.\int_0^2 x\,dx=\int_0^1 x\,dx+\int_1^2 x\,dx=\tfrac12+\tfrac32=2.
References
  • B. Yan, The Riemann–Stieltjes Integral, Definition 6.3 and Theorem 6.9(c), pp. 2 and 6–7 (specializing the integrator to γ(x)=x\gamma(x)=x): course notes PDF.