Sideway
output.to from Sideway
Calculus


IntegralRules of IntegrationApplications of Integration


GeometricalPhysical

Curve Length & PerimeterArea of Surface of Revolution

Draft for Information Only

Content

Approximation of Accumulative Physical Quantity
  Area of Surface of Revolution
  Lateral Surface Areas of Common Shapes
   Lateral Surface Area of Cylinder
   Lateral Surface Area of Cone
   Lateral Surface Area of Conical Frustum
   Surface Area of Sphere

Approximation of Accumulative Physical Quantity

Similarly to the sum of accumulative physical quantity, physical quantity related to the length of function can also be determine by integration when the physical qunatity can be expressed as infinitesimal elements.

 IMAGE...

Area of Surface of Revolution

Similar to solid of revolution, a surface of revolution is formed by rotating a curve about an axis. In solid of revolution, the infinitesimal element is approximated by πy2Δx. And the infinitesimal element can also be determined in the same way.

The most simple surface of revolution is the circular cylinderical surface by rotating a horizontal line. The lateral surface area of a circular cylinder can be considered as the extrusion of the circular perimeter along the length of the cylinder. The flat pattern is a rectangular sheet and is equal to 2πrL. Graphically

 IMAGE...

When the rotating curve is a sloped line with one end on the axis of rotation, the surface of revolution is a right conical surface. Since the slant height is the same for a right lateral conical surface, the flat pattern is a circular sector with radius of the slant height and the arc length of the circular sector is equal to the circumference of the cone base.  Graphically

 IMAGE...

Mathematically, for the right conical surface, the lateral surface area is

 IMAGE...

By making use of similar triangles. The lateral surface of a conical frustum is

 IMAGE...

An infinitesimal lateral surface element is similar to the lateral surface of a conical frustum. Imply

 IMAGE...

Lateral Surface Areas of Common Shapes

Lateral Surface Area of Cylinder

 IMAGE...

Lateral surface area of cylinder by integration is

 IMAGE...

Lateral Surface Area of Cone

 IMAGE...

Lateral surface area of cone by integration is

 IMAGE...

Lateral Surface Area of Conical Frustum

 IMAGE...

Lateral surface area of conical frustum by integration is

 IMAGE...

Surface Area of Sphere

 IMAGE...

Surface area of sphere by integration is

 IMAGE...

©sideway

ID: 120100018 Last Updated: 1/14/2012 Revision: 0 Ref:

close

References

  1. S. James, 1999, Calculus
  2. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science & Engineering
close

Latest Updated LinksValid XHTML 1.0 Transitional Valid CSS!Nu Html Checker Firefox53 Chromena IExplorerna
IMAGE

Home 5

Business

Management

HBR 3

Information

Recreation

Hobbies 8

Culture

Chinese 1097

English 339

Reference 79

Computer

Hardware 249

Software

Application 213

Digitization 32

Latex 52

Manim 205

KB 1

Numeric 19

Programming

Web 289

Unicode 504

HTML 66

CSS 65

SVG 46

ASP.NET 270

OS 429

DeskTop 7

Python 72

Knowledge

Mathematics

Formulas 8

Algebra 84

Number Theory 206

Trigonometry 31

Geometry 34

Coordinate Geometry 2

Calculus 67

Complex Analysis 21

Engineering

Tables 8

Mechanical

Mechanics 1

Rigid Bodies

Statics 92

Dynamics 37

Fluid 5

Fluid Kinematics 5

Control

Process Control 1

Acoustics 19

FiniteElement 2

Natural Sciences

Matter 1

Electric 27

Biology 1

Geography 1


Copyright © 2000-2024 Sideway . All rights reserved Disclaimers last modified on 06 September 2019