There are some problems in engineering and science that require evaluation of the volume of a geometric shape as it passes within another curve or shape along the 'x' or 'y' axis. Such examples include a square shape that intersects with a circle, circle whose diameter is bounded by linear functions, etc....
In all these cases, we must envision a 3-dimensional view of a shape that is sliding along the 'x' or 'y' axis. The volume is calculated as the sum of the areas of the shape as it 'slides' along each point of the 'x' (or 'y') axis.
If the shape slides along the 'x' axis, the volume function integrates over each point of the 'x' axis within the bounds of the functions:
$V={\int }_{a}^{b}A\left(x\right)dx$
If the shape slides along the 'y' axis, the function integrates over each point of the 'y' axis within the bounds of the functions:
$V={\int }_{a}^{b}A\left(y\right)dy$
In this instance, we will calculate the volume of a shape that is formed by a square that passes along the 'y' axis; one of the sides of the square is inscribed within a circle of the formula x 2+ y 2= 16.
Volume of Shapes Bounded by Shapes