Definition of Negative Integral - Apostol Calculus Vol.1

Question

I am going through Apostol's Calculus. In the definition of integral's properties it is said that:

$ \int_a^b s(x) \,dx = - \int_b^a s(x) \,dx $

So do I have a negative Area (impossible)? What's the meaning of this? Don't these two integrals definitions represent the same Area?

Answer 1

If $f|_{[a,b]}$ is Riemann-integrable and such that $(\forall x\in[a,b]):f(x)\geqslant0$, then $\int_a^bf(x)\,\mathrm dx$ means the are below the graph of $f$ (and above the $x$-axis). Then, if $c\in[a,b]$, we have$$\int_a^bf(x)\,\mathrm dx=\int_a^cf(x)\,\mathrm dx+\int_c^bf(x)\,\mathrm dx.\tag1$$The definition made in Apostol's textbook is done so that $(1)$ always holds, even if $c<a$ or that $c>b$. It has nothing to do with the computation of areas.