Question on Riemann Integrals

The upper Riemann integral is defined as $\int_a^{-b}f(x)dx = \inf U(P,f)$, and the lower Riemann integral is defined as $\int_{-a}^{b}f(x)dx = \sup L(P,f)$, where $U$ and $L$ denote the upper and lower sums respectively, and the supremum and infimum are taken over all partitions $P$ of $[a,b]$.

I am able to visualize that $\sup L(P,f)$ and $\inf U(P,f)$ are basically the lower and upper Riemann sums as the number of 'rectangular blocks' tends to infinity, but I do not understand why the above two equalities hold. $\sup L(P,f)$ and $\inf U(P,f)$, I think, are essentially $\int_a^{b}f(x)dx$. But how could we say that this integral equals $\int_a^{-b}f(x)dx $ or $\int_{-a}^{b}f(x)dx$?

For example, let $f(x)=x^2, a = 1$ and $b=2$. In this case $\int_a^{b}f(x)dx$ = $\frac{7}{3}$ and $\int_{-a}^bf(x)dx = 3$ .

We can say that $\int_a^bf(x)\,\mathrm dx$ is equal to both ${\underline\int}_a^bf(x)\,\mathrm dx$ and $\overline{\int}_a^bf(x)\,\mathrm dx$ because that's how we define $\int_a^bf(x)\,\mathrm dx$: it is equal to both those numbers when they're equal.

Besides,$$\underline{\int}_1^2x^2\,\mathrm dx=\frac73\ne3.$$It's up to you to tell us how is it that you got $3$.