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 $[a,b]$ .

$$S_P=\sum _{i=1}^{n}f(x_i)\mathrm{\Delta }{x}_i$$

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 $F^{\prime }(x)\equiv f(x)$. 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.

1. Partition $[a,b]$ into subintervals, i.e.

$$P:a=x_0<x_1<\cdots <x_n=b$$

1. 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 $\mathrm{\Delta }{x}_k.$ The volume for this slab is approximately the same as the volume of cylindrical solid

$$\text{Volume of the }k\text{-th slab}\approx V_k=A(x_k)\mathrm{\Delta }{x}_k$$

1. The volume $V$ of the entire solid $S$ is approximated by the sum of cylindrical volumes

$$V\approx \sum _{k=1}^n{V}_k=\sum _{k=1}^{n}A(x_k)\mathrm{\Delta }{x}_k$$

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

1. Sketch the solid and a typical cross-section
2. Find the formula of the area $A(x)$ for general cross-section
3. Find the upper and lower limits of integration
4. 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.

$$V={\int }_0^{3}A(x)dx={\int }_0^3{x}^{2}dx={\frac{x^3}{3}]}_0^3=9.$$

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)