Find the volume of a shape whose base is defined as the area between graphs (for e.g. the area between $\sin x$ and $\cos x$) and whose height is defined as another function (e.g. $\cos x$)? This graphic could help you. The ||'s in the end means that the top edges are parallel.
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With those conditions you could find the volume without knowing anything about double or triple integrals by just using intuition. So if I understood it well, you just have a $f(x)$ function on the xy-plane and you "drag" it out to fill the space down the z-axis. Then the volume is surface on xy-plane times the length on z-axis.
$$V = S_{xy} \cdot z = z \int_a^b f(x)dx$$
Where $a$ and $b$ are coordinates on x-axis.
So you have a region $D$ in the $xy$-plane described as the area between two functions $y = l(x)$ and $y = u(x)$ for $x \in [a,b]$. I.e., $$D = \{(x,y) \in \Bbb R^2 \mid a\le x \le b\text{ and }l(x) \le y \le u(x)\}$$
And you are also given a function $f(x,y)$ such that $f(x,y) \ge 0$ for $(x,y) \in D$, and are looking for the volume of the solid $$E = \{(x,y,z) \in \Bbb R^3 \mid (x,y) \in D\text{ and } 0 \le z \le f(x,y)\}$$
This can be determined by the iterated integral $$\begin{align}V &= \iiint_E 1\,dV \\&= \int_a^b\int_{l(x)}^{u(x)}\int_0^{f(x,y)} 1\,dz\,dy\,dx\\&=\int_a^b\int_{l(x)}^{u(x)}f(x,y)\,dy\,dx\end{align}$$