Proof of Various Integral Properties — Question 3

Question 3 - Solution

We use the definition of the definite integral in terms of Riemann sums.

Consider the interval $[a,a]$. Any partition of this interval consists of a single point: $$a=x_0=x_1.$$

The width of the subinterval is $$\Delta x = x_1-x_0 = a-a = 0.$$

Choose any sample point $x^*$ in the interval $[a,a]$.

A Riemann sum for $f$ over $[a,a]$ is $$\sum f(x^*)\,\Delta x = f(x^*)\cdot 0 = 0.$$

Since every Riemann sum over $[a,a]$ equals $0$, the limit of the Riemann sums exists and equals $0$.

By definition of the definite integral, $$\int_a^a f(x)\,dx = 0.$$

$$\boxed{\int_a^a f(x)\,dx = 0}$$