Question

Prove that for the region between the graph of a function f and the x-axis on an interval [a, b], the absolute area is always greater than or equal to the signed area.

Short answer

Hence, proved.

Step-by-step solution

Step 1. Given Information.

The region is between the graph of a function f and the x-axis on an interval [a, b]

Step 2. Signed area and absolute area.

$$\begin{gathered} Signedarea:{\int }_a^{b}f\left(x\right)dx. \\ Absolutearea:{\int }_a^b\left|f\left(x\right)\right|dx. \end{gathered}$$

Step 3. Proof.

$$\begin{gathered} Asweallknow: \\ \left|x\right|⩾x, \\ So, \\ f\left(\left|x\right|\right)⩾f\left(x\right), \\ Therefore,weget, \\ {\int }_a^b\left|f\left(x\right)\right|dx⩾{\int }_a^{b}f\left(x\right)dx. \\ Hence,proved. \end{gathered}$$

This proves that the absolute area is always greater than or equal to the signed area.