Proof of Various Integral Properties — Question 5

Question 5 - Solution

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

Let $$a=x_0<x_1<\cdots<x_n=b$$ be a partition of the interval $[a,b]$, and let $$\Delta x_i = x_i-x_{i-1}.$$

Choose a sample point $x_i^*$ in each subinterval $[x_{i-1},x_i]$.

A Riemann sum for $f$ on $[a,b]$ is $$\sum_{i=1}^n f(x_i^*)\,\Delta x_i.$$

Since $f(x)\ge 0$ for all $x\in[a,b]$ and $\Delta x_i>0$, each term in the sum satisfies $$f(x_i^*)\,\Delta x_i \ge 0.$$

Thus, every Riemann sum for $f$ over $[a,b]$ is nonnegative: $$\sum_{i=1}^n f(x_i^*)\,\Delta x_i \ge 0.$$

Now take the limit as the norm of the partition $\|P\|\to 0$. Since $f$ is integrable, the limit of the Riemann sums exists and equals the definite integral: $$\int_a^b f(x)\,dx = \lim_{\|P\|\to 0}\sum_{i=1}^n f(x_i^*)\,\Delta x_i.$$

Because every sum in the limit is nonnegative, the limit itself must be nonnegative.

Therefore, $$\int_a^b f(x)\,dx \ge 0.$$

$$\boxed{\int_a^b f(x)\,dx \ge 0}$$