Chap6_Sec3

# Chap6_Sec3 - 6 APPLICATIONS OF INTEGRATION APPLICATIONS...

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

APPLICATIONS OF INTEGRATION APPLICATIONS OF INTEGRATION 6

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

View Full Document
6.3 Volumes by Cylindrical Shells APPLICATIONS OF INTEGRATION In this section, we will learn: How to apply the method of cylindrical shells to find out the volume of a solid.
Some volume problems are very difficult to handle by the methods discussed in Section 6.2 VOLUMES BY CYLINDRICAL SHELLS

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

View Full Document
Let’s consider the problem of finding the volume of the solid obtained by rotating about the y -axis the region bounded by y = 2 x 2 - x 3 and y = 0. VOLUMES BY CYLINDRICAL SHELLS
If we slice perpendicular to the y -axis, we get a washer. However, to compute the inner radius and the outer radius of the washer, we would have to solve the cubic equation y = 2 x 2 - x 3 for x in terms of y . That’s not easy. VOLUMES BY CYLINDRICAL SHELLS

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

View Full Document
Fortunately, there is a method—the method of cylindrical shells—that is easier to use in such a case. VOLUMES BY CYLINDRICAL SHELLS
The figure shows a cylindrical shell with inner radius r 1 , outer radius r 2 , and height h . CYLINDRICAL SHELLS METHOD

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

View Full Document
Its volume V is calculated by subtracting the volume V 1 of the inner cylinder from the volume of the outer cylinder V 2 . CYLINDRICAL SHELLS METHOD
Thus, we have: 2 1 2 2 2 1 2 2 2 1 2 1 2 1 2 1 2 1 ( ) ( )( ) 2 ( ) 2 V V V r h r h r r h r r r r h r r h r r π π π π π = - = - = - = + - + = - CYLINDRICAL SHELLS METHOD

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

View Full Document
Let ∆ r = r 2 r 1 (thickness of the shell) and (average radius of the shell). Then, this formula for the volume of a cylindrical shell becomes: 2 V rh r π = Formula 1 ( 29 1 2 1 2 r r r = + CYLINDRICAL SHELLS METHOD
The equation can be remembered as: V = [circumference] [height] [thickness] CYLINDRICAL SHELLS METHOD 2 V rh r π =

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.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern