Main idea: To find the volume of
a solid, we slice the solid into thin pieces and add up the
volumes of the pieces.

When we slice a solid, we get cross-sections
of the solid. For example, when we slice a solid cylinder,
we see that we can get cross-sections that are rectangles or
circles, depending upon how we slice.

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In general, we do not have nice solids like a cylinder, but we
can slice any solid. We put our solid on axes, so that we
can coordinatize our slices, as in the diagrams below.
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We slice the solid into these cross-sectional pieces, then add up the volumes of each of the pieces, to
get the volume of the solid. If a slice has
cross-sectional area $A(x)$ and thickness $\Delta x$ at each
particular $x$-value, then the slice has approximate volume $A(x)
\Delta x$. Adding up the volumes of the slices and taking a limit
as we slice finer and finer, we get

$$ \hbox{Volume}
= \int_a^b A(x)\, dx$$

where $a$ and $b$ are the $x$-values of the leftmost and
rightmost slices.

Of course, we could also slice horizontally,

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http://tutorial.math.lamar.edu/
in which case we would find the volume to be $\int_c^d A(y)\,dy$,
where $c$ is the smallest $y$-value, and $d$ is the largest
($c=0,d=h$ in the second diagram above).