Presentation is loading. Please wait.

Presentation is loading. Please wait.

MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.3 – Volumes by Cylindrical Shells Copyright © 2006 by Ron Wallace,

Published by Modified over 8 years ago

Similar presentations

Presentation on theme: "MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.3 – Volumes by Cylindrical Shells Copyright © 2006 by Ron Wallace,"— Presentation transcript:

1 MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.3 – Volumes by Cylindrical Shells Copyright © 2006 by Ron Wallace, all rights reserved.

2 Reminder: Definition of a Definite Integral Simplified Version where …

3 Cylindrical Shells h r t If t is small …

4 f(x) ba Volumes Using Cylindrical Shells xkxk radius of cylinder height of cylinder circumference of cylinder thickness of cylinder Cross-section is taken parallel to the line of rotation.

5 Solids of Revolution - Example A solid obtained by revolving a region around a line. Find the volume of a cone of height h and bottom radius r. r h NOTE: Cross-section is parallel to the axis of rotation.

6 Volumes of Solids of Revolution Comparing Methods Disks & WashersCylindrical Shells Cross-section: Perpendicular to the line of rotation. Cross-section: Parallel to the line of rotation. r is the length of the cross- section in terms of x or y. Rotate about the x-axis: Rotate about the y-axis: h is the length of the cross- section in terms of x or y. Rotate about the x-axis: Rotate about the y-axis:

Download ppt "MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.3 – Volumes by Cylindrical Shells Copyright © 2006 by Ron Wallace,"

Similar presentations

Volume by Parallel Cross Section; Disks and Washers

Volume by Parallel Cross Section; Disks and Washers
Volumes. Right Cylinder Volume of a Right Cylinder (Slices) Cross section is a right circular cylinder with volume Also obtained as a solid of revolution.

Volumes. Right Cylinder Volume of a Right Cylinder (Slices) Cross section is a right circular cylinder with volume Also obtained as a solid of revolution.
More on Volumes & Average Function Value. Average On the last test (2), the average of the test was: FYI - there were 35 who scored a 9 or 10, which means.

More on Volumes & Average Function Value. Average On the last test (2), the average of the test was: FYI - there were 35 who scored a 9 or 10, which means.
VOLUMES Volume = Area of the base X height. VOLUMES.

VOLUMES Volume = Area of the base X height. VOLUMES.
 A k = area of k th rectangle,  f(c k ) – g(c k ) = height,  x k = width. 6.1 Area between two curves.

 A k = area of k th rectangle,  f(c k ) – g(c k ) = height,  x k = width. 6.1 Area between two curves.
Applications of Integration Copyright © Cengage Learning. All rights reserved.

Applications of Integration Copyright © Cengage Learning. All rights reserved.
APPLICATIONS OF INTEGRATION 6. 6.3 Volumes by Cylindrical Shells APPLICATIONS OF INTEGRATION In this section, we will learn: How to apply the method of.

APPLICATIONS OF INTEGRATION Volumes by Cylindrical Shells APPLICATIONS OF INTEGRATION In this section, we will learn: How to apply the method of.
Section 6.2: Volumes Practice HW from Stewart Textbook (not to hand in) p. 457 # 1-13 odd.

Section 6.2: Volumes Practice HW from Stewart Textbook (not to hand in) p. 457 # 1-13 odd.
MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.2 – Volumes by Slicing; Disks and Washers Copyright © 2006 by Ron.

MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.2 – Volumes by Slicing; Disks and Washers Copyright © 2006 by Ron.
Lesson 6-2c Volumes Using Washers. Ice Breaker Volume = ∫ π(15 - 8x² + x 4 ) dx x = 0 x = √3 = π ∫ (15 - 8x² + x 4 ) dx = π (15x – (8/3)x 3 + (1/5)x 5.

Lesson 6-2c Volumes Using Washers. Ice Breaker Volume = ∫ π(15 - 8x² + x 4 ) dx x = 0 x = √3 = π ∫ (15 - 8x² + x 4 ) dx = π (15x – (8/3)x 3 + (1/5)x 5.
6.2 - Volumes. Definition: Right Cylinder Let B 1 and B 2 be two congruent bases. A cylinder is the points on the line segments perpendicular to the bases.

6.2 - Volumes. Definition: Right Cylinder Let B 1 and B 2 be two congruent bases. A cylinder is the points on the line segments perpendicular to the bases.
4/30/2015 Perkins AP Calculus AB Day 4 Section 7.2.

4/30/2015 Perkins AP Calculus AB Day 4 Section 7.2.
APPLICATIONS OF INTEGRATION 6. 6.3 Volumes by Cylindrical Shells APPLICATIONS OF INTEGRATION In this section, we will learn: How to apply the method of.

APPLICATIONS OF INTEGRATION Volumes by Cylindrical Shells APPLICATIONS OF INTEGRATION In this section, we will learn: How to apply the method of.
Section 6.1 Volumes By Slicing and Rotation About an Axis

Section 6.1 Volumes By Slicing and Rotation About an Axis
Volume: The Disk Method

Volume: The Disk Method
TOPIC APPLICATIONS VOLUME BY INTEGRATION. define what a solid of revolution is decide which method will best determine the volume of the solid apply the.

TOPIC APPLICATIONS VOLUME BY INTEGRATION. define what a solid of revolution is decide which method will best determine the volume of the solid apply the.
Chapter 6 – Applications of Integration

Chapter 6 – Applications of Integration
 Find the volume of y= X^2, y=4 revolved around the x-axis  Cross sections are circular washers  Thickness of the washer is xsub2-xsub1  Step 1) Find.

 Find the volume of y= X^2, y=4 revolved around the x-axis  Cross sections are circular washers  Thickness of the washer is xsub2-xsub1  Step 1) Find.
Section 6.2.  Solids of Revolution – if a region in the plane is revolved about a line “line-axis of revolution”  Simplest Solid – right circular cylinder.

Section 6.2.  Solids of Revolution – if a region in the plane is revolved about a line “line-axis of revolution”  Simplest Solid – right circular cylinder.
SECTION 7.3 Volume. VOLUMES OF AN OBJECT WITH A KNOWN CROSS-SECTION  Think of the formula for the volume of a prism: V = Bh.  The base is a cross-section.

SECTION 7.3 Volume. VOLUMES OF AN OBJECT WITH A KNOWN CROSS-SECTION  Think of the formula for the volume of a prism: V = Bh.  The base is a cross-section.

Similar presentations

Ads by Google