Proof of Various Integral Properties — Question 10

Question 10 - Solution

Define the function $$h(x)=M-f(x).$$

Since $f$ is integrable on $[a,b]$ and $M$ is a constant, the function $h$ is also integrable on $[a,b]$.

From the assumption $f(x)\le M$ for all $x\in[a,b]$, we have $$h(x)=M-f(x)\ge 0 \quad\text{for all }x\in[a,b].$$

By the positivity property of integrals, $$\int_a^b h(x)\,dx \ge 0.$$

Substitute back for $h(x)$: $$\int_a^b \bigl(M-f(x)\bigr)\,dx \ge 0.$$

Use linearity of the integral: $$\int_a^b M\,dx - \int_a^b f(x)\,dx \ge 0.$$

Since $$\int_a^b M\,dx = M(b-a),$$ we obtain $$M(b-a)-\int_a^b f(x)\,dx \ge 0.$$

Rearranging gives $$\int_a^b f(x)\,dx \le M(b-a).$$

$$\boxed{\int_a^b f(x)\,dx \le M(b-a)}$$