Properties of Riemann integral $\left| \int_b^{a}f\right| \lt \int_b^{a}|f|$？

Question

$$f(x) \lt |f(x)|\implies\int_b^{a}f \lt \int_b^{a}|f|,$$

$$-f(x) \lt |f(x)|\implies-\int_b^{a}f \lt \int_b^{a}|f|.$$

Hence , $$-\int_b^{a}|f| \lt \int_b^{a}f \lt \int_b^{a}|f|.$$

Therefore $$\left| \int_b^{a}f\right| \lt \int_b^{a}|f|.$$

This inequality is correct? My book has a typo.

Answer 1

Yes it is correct. Why do you think it is not? It's important to tell us what you're doubting so that we can explain.

The proof is correct as well.