Volume of Solid of Revolution

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Theorem

Let $f: \R \to \R$ be a real function which is integrable on the interval $\closedint a b$.

Let the points be defined:

$A = \tuple {a, \map f a}$
$B = \tuple {b, \map f b}$
$C = \tuple {b, 0}$
$D = \tuple {a, 0}$

Let the figure $ABCD$ be defined as being bounded by the straight lines $y = 0$, $x = a$, $x = b$ and the curve defined by $\set {\map f x: a \le x \le b}$.

Let the solid of revolution $S$ be generated by rotating $ABCD$ around the $x$-axis (that is, $y = 0$).


Then the volume $V$ of $S$ is given by:

$\ds V = \pi \int_a^b \paren {\map f x}^2 \rd x$


Parametric Form

Let $x: \R \to \R$ and $y: \R \to \R$ be real functions defined on the interval $\closedint a b$.

Let $y$ be integrable on the (closed) interval $\closedint a b$.

Let $x$ be differentiable on the (open) interval $\openint a b$.

Let the points be defined:

$A = \tuple {\map x a, \map y a}$
$B = \tuple {\map x b, \map y b}$
$C = \tuple {\map x b, 0}$
$D = \tuple {\map x a, 0}$

Let the figure $ABCD$ be defined as being bounded by the straight lines $y = 0$, $x = a$, $x = b$ and the curve defined by:

$\set {\tuple {\map x t, \map y t}: a \le t \le b}$

Let the solid of revolution $S$ be generated by rotating $ABCD$ around the $x$-axis (that is, $y = 0$).


Then the volume $V$ of $S$ is given by:

$\ds V = \pi \int_a^b \paren {\map y t}^2 \map {x'} t \rd t$


Proof

VolumeOfSolidOfRevolution.png


Consider a rectangle bounded by the lines:

$y = 0$
$x = \xi$
$x = \xi + \delta x$
$y = \map f x$

Consider the right circular cylinder generated by revolving it about the $x$-axis.

By Volume of Right Circular Cylinder, the volume of this cylinder is:

$V_\xi = \pi \paren {\map f x}^2 \delta x$


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Historical Note

The technique of finding the Volume of Solid of Revolution by dividing up the solid of revolution into many thin disks and approximating them to cylinders was devised by Johannes Kepler sometime around or after $1612$, reportedly on the occasion of his wedding in $1613$.

His inspiration was in the problem of finding the volume of wine barrels accurately.

He published his technique in his $1615$ work Nova Stereometria Doliorum Vinariorum (New Stereometry of Wine Barrels).

Gottfried Wilhelm von Leibniz redefined the problem by applying the techniques of integral calculus around $1680$.


Sources

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