Chap8_Sec2

# Chap8_Sec2 - 8 FURTHER APPLICATIONS OF INTEGRATION OF...

This preview shows pages 1–12. Sign up to view the full content.

FURTHER APPLICATIONS FURTHER APPLICATIONS OF INTEGRATION OF INTEGRATION 8

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
8.2 Area of a Surface of Revolution In this section, we will learn about: The area of a surface curved out by a revolving arc. FURTHER APPLICATIONS OF INTEGRATION
SURFACE OF REVOLUTION A surface of revolution is formed when a curve is rotated about a line. Such a surface is the lateral boundary of a solid of revolution of the type discussed in Sections 6.2 and 6.3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
AREA OF A SURFACE OF REVOLUTION We want to define the area of a surface of revolution in such a way that it corresponds to our intuition. If the surface area is A , we can imagine that painting the surface would require the same amount of paint as does a flat region with area A .
Let’s start with some simple surfaces. AREA OF A SURFACE OF REVOLUTION

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
CIRCULAR CYLINDERS The lateral surface area of a circular cylinder with radius r and height h is taken to be: A = 2 πrh We can imagine cutting the cylinder and unrolling it to obtain a rectangle with dimensions of 2 πrh and h .
CIRCULAR CONES We can take a circular cone with base radius r and slant height l , cut it along the dashed line as shown, and flatten it to form a sector of a circle with radius and central angle θ = 2 πr / l .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
CIRCULAR CONES We know that, in general, the area of a sector of a circle with radius l and angle θ is ½ l 2 θ .
CIRCULAR CONES So, the area is: Thus, we define the lateral surface area of a cone to be A = πr l. 2 2 1 1 2 2 2 r A l l rl l π θ = = = ÷

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
AREA OF A SURFACE OF REVOLUTION What about more complicated surfaces of revolution?
AREA OF A SURFACE OF REVOLUTION If we follow the strategy we used with arc length, we can approximate the original curve by a polygon. When this is rotated about an axis, it creates

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 01/06/2012 for the course MATH 2414.S01 taught by Professor Alans.grave during the Fall '11 term at Collins.

### Page1 / 46

Chap8_Sec2 - 8 FURTHER APPLICATIONS OF INTEGRATION OF...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online