Volume of Revolution

The function $y = x^3 - x$rotated about the $x$-axis

Asolid of revolutionis a three-dimensional object obtained by rotating a function in the plane about a line in the plane. The volume of this solid may be calculated by means ofintegration. Common methods for finding the volume are thedisc method,shell method, andPappus's centroid theorem.

Volumes of revolution are useful for topics in engineering, medical imaging, andgeometry. The manufacturing of machine parts and the creation of MRI images both require understanding of these solids.

Main Article:Disc Method

Thedisc methodcalculates the volume of the full solid of revolution by summing the volumes of thin vertical circular disks. This is similar to the notion of integration as beingthe sum of an infinite number of rectangles. The disc method imagines the solid of revolution as a stack of discs of varying radii. It gives rise to the formula for rotation of the region bounded by $y = f(x)$, $y = 0$, $x = a$, and $x = b$about the $x$-axis:

$V = \int_a^b \pi (f(x))^2 \, dx.$

注意,该地区可能是一个旋转arbitrary line $y = c$that does not intersect the region with the formula

$V = \left| \int_a^b \pi (f(x) - c)^2 \, dx - \int_a^b \pi c^2 \, dx \right|.$

Main Article:Shell Method

Theshell methodcalculates the volume of the full solid of revolution by summing the volumes of thin cylindrical shells. This gives rise to the formula for rotation of the region bounded by $y = f(x)$, $y = 0$, $x = a$, and $x = b$about the $y$-axis:

$V = \int_a^b 2 \pi x f(x) \, dx.$

注意,该地区可能是一个旋转arbitrary line $x = c$that does not intersect the region with the formula

$V = \int_a^b 2 \pi \lvert x - c \rvert f(x) \, dx.$

When the function $f(x)$does not have an inverse with an easily expressibleantiderivative,n the disc method should be used for rotations around horizontal lines and the shell method for rotations around vertical lines. In general, the method that gives the simpler integral is preferred.

Determine the volume of the solid obtained by rotating the region bounded by $y = \frac{1 - x}{1 + x^2}$and the first quadrant about the $y$-axis.

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The function rotated about the $y$-axis

The region in question is bounded by $y = \frac{1 - x}{1 + x^2}$, $y = 0$, $x = 0$, and $x = 1$. Since $f(x) = \frac{1 - x}{1 + x^2}$probably has a somewhat messy inverse, there is no reason not to use the shell method (for rotation about a vertical line, the $y$-axis). The volume in question is then

$\begin{aligned} \int_0^1 2\pi x \cdot \frac{1 - x}{1 + x^2} \, dx &= 2\pi \int_0^1 \left(\frac{x}{1 + x^2} + \frac{1}{1 + x^2} - 1\right) \, dx \\ \\ &= 2\pi \big(\ln\sqrt{2} + \tfrac{\pi}{4} - 1\big).\ _\square \end{aligned}$

Suppose the region bounded by $y = f(x)$, $y = g(x)$, $x = a$, and $x = b$is rotated over a line. Suppose further that $f(x) \ge g(x)$for all $x \in [a, \, b]$.

If the line is vertical $x = c$(where $c \not\in (a, \, b)$), then the shell method yields a volume

$V = \int_a^b 2 \pi \lvert x - c \rvert \big(f(x) - g(x)\big) \, dx.$

If the line is horizontal $y = c$(where $c \not\in (a, \, b)$), then the disc method yields a volume

$V = \int_a^b \pi \Big\lvert \big(f(x) - c\big)^2 - \big(g(x) - c\big)^2 \Big\rvert \, dx.$

• Arc Length
• Disc Method
• Shell Method