周次: 1 日期: February 25, 2022 节次: 2
Outline
For solids of revolution, the volume is relatively easy to find out. In section 6.1 and 6.2, we developed three method for finding volumes. The first two are based on the area of cross-sections, therefore these cases are direct applications of the formula $V = \int_a^b A(x) dx.$ The third case uses cylindrical shells for slicing, hence the formula is different. In this lecture, we look at some examples.
Volume by Slicing — Disk Method
Example 4. A Solid of Revolution (About the $x$-axis)
The region between the curve $y=\sqrt{x}$, $0\le x\le 4$ about the $x$-axis.
Example 5. Volume of a sphere (omitted)
The circle $x^2+y^2 = a^2$ is rotated about the $x$-axis to generate a sphere. Find its volume.
Example 6. Rotation about a line $y=1$
A solid is generated by revolving the region bounded by $y=\sqrt{x}$ and the lines $y=1,$ $x=4$ about the line $y=1.$
Example 7. Rotation about the $y$-axis
Find the volume by revolving the region between the $y$ -axis and the curve $x=2/y$ , $1\le y\le 4$ , about the $y$-axis.
Example 8. Rotation About a Vertical Axis (omitted)
Find the volume of the solid generated by revolving the region between the parabola $x=y^2+1$ and the line $x=3$ about the line $x=3.$
Volume by Slicing — Washer Method
Suppose we are revolving a region about the $x$-axis. The region is bounded by two curves $y=R(x)$ and $y=r(x)$ (where $R(x)\ge r(x)\ge 0$ ), then the cross-section of the solid is an annual region. By using the general formula $V = \int_a^b A(x) dx$, the volume is given by
$$ V = \int_a^b A(x) dx = \pi\int_a^b(R(x)^2-r(x)^2)dx. $$
Since the cross-section is an annual shape, the method is called the washer method.
Example 9. A Washer Cross-Section (Rotation about the $x$-axis)
The region bounded by the curve $y=x^2+1$ and the line $y=-x+3$ is revolved about the $x$-axis to generate a solid.
Example 11. A Washer Cross-Section ( Rotation about the $y$-axis) (Omitted)
The region bounded by the parabola $y=x^2$ and the line $y=2x$ in the first quadrant is revolved about the $y$-axis to generate a solid.
Volume by Cylindrical Shells
Instead of slicing a solid into slabs, we can slice a solid of revolution into cylindrical shells. This is useful when revolving a region defined by a non-invertible function $y=f(x)$ about $y$-axis (or axis parallel to $y$-axis).
Slicing A Solid of Revolution by Cylindrical Shells
Suppose we are revolving a plane region defined by $y=f(x)$ with $x\in[a,b]$ about an axis $x=L$ (assume $a\ge L$). Let
$$ P:a=x_0<x_1<x_2<\cdots<x_n = b $$
be a partition of the interval $[a,b]$ , let $c_k$ be the midpoint of $k$th subinterval $[x_{k-1},x_k]$. We approximate the region with rectangles based on the partition $P$ and height equals to $f(c_k)$.
If these rectangle is rotated about the vertical line $x=L$, then a cylindrical shell is swept out. The partition $P$ gives a series of cylindrical shells, whose volumes are given by (Why?)
$$ \Delta V_k = 2\pi\times \text{average shell radius}\times \text{shell height}\times \text{thickness}\ = 2\pi\times (c_k-L)\times f(c_k)\times\Delta x_k. $$
Then the volume for the solid $S$ is approximately
$$ V\approx \sum_{k=1}^n \Delta V_k $$
The limit of this Riemann sum as $||P||\to 0$ gives the volume of $S$
$$ V = \int_a^b 2\pi(x-L)f(x) dx\=\int_a^b 2\pi \times \text{(shell radius)}\times\text{(shell height)}\times dx. $$
Here $x$ is understand as thickness variable.
Example 1. Finding a Volume Using Shells
The region enclosed by the $x$-axis and the parabola $y=f(x)=3x-x^2$ is revolved about the vertical line $x=-1$ to generate the shape of a solid. Find the volume.
$$ V = \int_0^3 2\pi (x+1) (3x-x^2)dx = \frac{45\pi}{2}. $$
Example 2. Cylindrical Shells Revolving About the $y$-Axis. (Omitted)
The region bounded by the curve $y=\sqrt{x}$, the $x$-axis, and the line $x=4$ is revolved about the $y$-axis to generate a solid. Find the volume of the solid.
Example 3. Cylindrical Shells Revolving About the $x$-Axis
The region bounded by the curve $y=\sqrt{x}$, the $x$-axis, and the line $x=4$ is revolved about the $x$-axis to generate a solid. Find the volume of the solid.
Summary of the Shell Method
- Draw the region and sketch a line segment across it parallel to the axis of revolution. Label the segment’s height or length and distance from the axis of revolution (shell radius)
- Find the limits of integration for the thickness variable
- Integrate according to the formula.