W1-S1 Volumes by Slicing and Rotation About an Axis
周次: 1 日期: February 23, 2022 节次: 1
In this semester, we will be beginning with the applications of definite integrals. Today’s course is the contents in section 6.1 (Volumes by Slicing and Rotation About an Axis).
Outline
Review: Riemann sums & Definite Integral
Definite integral is dealing with summations of infinite many tiny quantities. For example, to calculate the area of region under the curve $y= f(x)$ .
The approximate “area” is given by Riemann sums associating with a partition $P$ of the finite closed interval $[a,b]$ .
$$ S_P = \sum_{i=1}^n f(x_i) \Delta x_i $$
This is a finite sum, and each term has a finite (not infinitely small) area.
To define the “area” precisely, we take the limit of $S_P$ as the norm of the partition $||P||$ approaches zero. The number we get is the following, called the definite integral of $f(x)$.
$$ \int_a^b f(x)dx. $$
To evaluate this integral, we have to find the anti-derivative of $f(x)$, i.e. $F(x)$ with $F'(x)\equiv f(x)$. Then the fundamental theorem of calculus shows
$$ \int_a^b f(x)dx = F(b)-F(a). $$
Volumes by Slicing
In this section, we will apply this idea to the calculation of volumes. We can show that for certain regular solid ( shape), we can calculate the volume if the area for cross-sections can be known in prior.
Basic Knowledge For any cylindrical solid, if the base area is $A$ and height is $h$
, the volume is
$$ \text{Volume} = A \times h $$
For solids that are not in cylindrical shape, the volume can be found by the method of slicing.
Definition (Cross-section)
A cross-section of a solid $S$ is the plane region formed by intersecting $S$ with a plane. E.g. the following graph shows a solid $S$ has a cross-section with plane $P:{ x=x_0 }$
Method of Finding the Volumes by Slicing
Suppose a solid $S$ can be positioned between two parallel planes $x=a$ and $x=b$.
To find the volume $V$ of $S$, assuming we know the area for any cross-section of $S$ with any plane $P$ perpendicular to the $x$ -axis and positioned at $x$. The area for the cross-section will be denoted by $A(x)$.
Then to find the approximation for volume $V$, we will go through the following steps.
- Partition $[a,b]$ into subintervals, i.e.
$$ P: a = x_0 < x_1 < \cdots< x_n = b $$
-
The planes $P_{x_i}$ at $x=x_i$ slice $S$ into thin “slabs”.
These slabs are approximated by cylindrical solid with base area $A(x_k)$ (How to estimate the inaccuracy ?), and thickness $\Delta x_k.$ The volume for this slab is approximately the same as the volume of cylindrical solid
$$ \text{Volume of the }k\text{-th slab} \approx V_k = A(x_k)\Delta x_k $$
- The volume $V$ of the entire solid $S$ is approximated by the sum of cylindrical volumes
$$ V\approx \sum_{k=1}^n V_k = \sum_{k=1}^n A(x_k)\Delta x_k $$
This is a Riemann sum for the function $A(x)$ on $[a,b]$. Assuming $A(x)$ is a continuous function on $[a,b]$, then we can expect the approximation to improve as the norm $||P||\to 0$. (❓Which means the approximation is approaching the real value, why.)
Then the volume is defined by
$$ V = \int_a^b A(x) dx. $$
Definition. (Volume)
The volume of a solid with known integral equation cross-section area $A(x)$ from $x=a$ to $x=b$ is given by the integral
$$ V = \int_a^b A(x) dx. $$
Calculating the Volume of a Solid
- Sketch the solid and a typical cross-section
- Find the formula of the area $A(x)$ for general cross-section
- Find the upper and lower limits of integration
- Calculate the definite integral using the Fundamental Theorem.
Example 1. Volume of a Pyramid
A pyramid $3$ meter high has a square base that is $3$ meter on a side.
$$ V = \int_0^3 A(x)dx = \int_0^3 x^2 dx = \left.\frac{x^3}{3}\right]_{0}^3 = 9. $$
Example 2. Volume of a Wedge
A curved wedge is cut from a cylinder of radius 3 by two planes. The first plane is perpendicular to the axis of the cylinder. The second one crosses the first one at a $45^\circ$ angle at the center of the of the cylinder. What is the volume $V$ for the wedge ?
- What is the area of the cross section $A(x)=?$
Example 3. Volume of a Cone (why $1/3$ that of the cylinder?)
A cone is $h$ in height, and $R$ in radius for the base.
Cavalieri’s Principle (祖暅原理)
Two solids with equal altitudes and identical cross-sectional areas at each height have the same volume. (Why?)
Example 4. The hemisphere and the cylinder with a cone been removed
A hemisphere with $R$ in radius, the cylinder has radius $R$ and height $R$ , the cone is described as in Example 3, but with $h=R$. What is the relation for the volume of the two solids?
Solids of Revolution
The solid generated by rotating a plane region about an axis in its plane is called a solid of revolution.
In this case, the area of cross-section is usually easy to find. There are a lot of examples.
Solids of Revolution: The Disk Method ( Disks as Slabs)
Example 5. 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 6. Volume of a sphere (omitted)
A sphere with radius $R$.
Example 7. 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 8. 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 9. Rotation about a vertical axis
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 $x=3.$
Solids of Revolution: The Washer Method (Washers as Slabs)
Example 10. 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)
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. (Omitted)