Ms. Yoakum Calculus. Click one: Disk Method Cylindrical Shells Method Washer Method Back Click one:

Slides:

Advertisements

Similar presentations

Find the exact volume of the solid formed by revolving the region enclosed by the following functions around the x-axis. Specify whether you are using.

Advertisements

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 Volume = Area of the base X height. VOLUMES.

The Disk Method (7.2) April 17th, I. The Disk Method Def. If a region in the coordinate plane is revolved about a line, called the axis of revolution,

Section 6.2: Volumes Practice HW from Stewart Textbook (not to hand in) p. 457 # 1-13 odd.

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.

4/30/2015 Perkins AP Calculus AB Day 4 Section 7.2.

APPLICATIONS OF INTEGRATION Volumes by Cylindrical Shells APPLICATIONS OF INTEGRATION In this section, we will learn: How to apply the method of.

Find the volume when the region enclosed by y = x and y = x 2 is rotated about the x-axis? - creates a horn shaped cone - area of the cone will be the.

Volume: The Disk Method

Miss Battaglia AP Calculus. Rotate around the x-axis and find the volume of y= 1 / 2 x from 0 to 4.

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

Volume. Find the volume of the solid formed by revolving the region bounded by the graphs y = x 3 + x + 1, y = 1, and x = 1 about the line x = 2.

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.

S OLIDS OF R EVOLUTION 4-G. Disk method Find Volume – Disk Method Revolve about a horizontal axis Slice perpendicular to axis – slices vertical Integrate.

Integral calculus XII STANDARD MATHEMATICS. Evaluate: Adding (1) and (2) 2I = 3 I = 3/2.

Review: Volumes of Revolution. x y A 45 o wedge is cut from a cylinder of radius 3 as shown. Find the volume of the wedge. You could slice this wedge.

Volume: The Shell Method Lesson 7.3. Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal.

Section 7.2 Solids of Revolution. 1 st Day Solids with Known Cross Sections.

A solid of revolution is a solid obtained by rotating a region in the plane about an axis. The sphere and right circular cone are familiar examples of.

7.3 Volumes by Cylindrical Shells

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

Do Now: #10 on p.391 Cross section width: Cross section area: Volume:

Chapter 6 – Applications of Integration 6.3 Volumes by Cylindrical Shells 1Erickson.

Volumes of Revolution Disks and Washers

Finding Volumes Disk/Washer/Shell Chapter 6.2 & 6.3 February 27, 2007.

Volumes of Revolution The Shell Method Lesson 7.3.

+ C.2.5 – Volumes of Revolution Calculus - Santowski 12/5/2015Calculus - Santowski.

Applications of Integration Copyright © Cengage Learning. All rights reserved.

8.2 Area of a Surface of Revolution

Volumes Using Cross-Sections Solids of Revolution Solids Solids not generated by Revolution Examples: Classify the solids.

Volumes Lesson 6.2.

Aim: Shell Method for Finding Volume Course: Calculus Do Now: Aim: How do we find volume using the Shell Method? Find the volume of the solid that results.

6.3 Volumes of Revolution Tues Dec 15 Do Now Find the volume of the solid whose base is the region enclosed by y = x^2 and y = 3, and whose cross sections.

7.3.1 Volume by Disks and Washers I. Solids of Revolution A.) Def- If a region in the plane is revolved about a line in the plane, the resulting solid.

Volumes by Slicing. disk Find the Volume of revolution using the disk method washer Find the volume of revolution using the washer method shell Find the.

Volume: The Disk Method. Some examples of solids of revolution:

Lecture 1 – Volumes Area – the entire 2-D region was sliced into strips Before width(  x) was introduced, only dealing with length ab f(x) Volume – same.

8.1 Arc Length and Surface Area Thurs Feb 4 Do Now Find the volume of the solid created by revolving the region bounded by the x-axis, y-axis, and y =

Volumes by Cylindrical Shells. What is the volume of and y=0 revolved around about the y-axis ? - since its revolving about the y-axis, the equation needs.

By: Rossboss, Chase-face, and Danny “the Rock” Rodriguez.

Calculus April 11Volume: the Disk Method. Find the volume of the solid formed by revolving the region bounded by the graph of and the x-axis (0 < x

C YLINDRICAL SHELLS C ONTINUED 4-Jcont Comparison of Methods.

6.3 Volumes by Cylindrical Shells. Find the volume of the solid obtained by rotating the region bounded,, and about the y -axis. We can use the washer.

Calculus 6-R Unit 6 Applications of Integration Review Problems.

Volumes of Solids of Rotation: The Disc Method

Drill: Find the area in the 4 th quadrant bounded by y=e x -5.6; Calculator is Allowed! 1) Sketch 2) Highlight 3) X Values 4) Integrate X=? X=0 X=1.723.

7-2 SOLIDS OF REVOLUTION Rizzi – Calc BC. UM…WHAT?  A region rotated about an axis creates a solid of revolution  Visualization Visualization.

6.3 Volumes By Cylindrical Shells This is an alternate method for finding the volume of a solid of revolution that uses cylindrical shells. The strip is.

VOLUMES BY CYLINDRICAL SHELLS CylinderShellCylinder.

Lesson 49 – Volumes of Revolution

The Shell Method Section 7.3.

Finding Volumes Disk/Washer/Shell

Review of Area: Measuring a length.

AP Calculus Honors Ms. Olifer

Volumes of Solids of Revolution

9.4: Finding Volume using Washer Method

7.3 Volume: The Shell Method

Applications Of The Definite Integral

7.2A Volumes by Revolution: Disk/Washer Method

6.2 Volumes by Revolution: Disk/Washer Method

Solids not generated by Revolution

Integration Volumes of revolution.

6.2 Solids of Revolution-Disk Method Warm Up

Solids of Revolution PART DEUX: WaShErS Rita Korsunsky

Volumes of Revolution: Disk and Washer Method

Presentation transcript:

Ms. Yoakum Calculus

Click one:

Disk Method Cylindrical Shells Method Washer Method Back Click one:

Next

Volume = R Next

Determine the volume of the solid obtained by rotating the region bounded by,,, and the x-axis about the x-axis. Problem courtesy of: Next

r Volume = Determine the volume of the solid obtained by rotating the region bounded by,,, and the x-axis about the x-axis. Problem courtesy of: Done

Next

(Horizontal Axis of Revolution) r Volume = Next

Determine the volume of the solid obtained by rotating the portion of the region bounded by and that lies in the first quadrant about the y-axis. Problem courtesy of: Next

Determine the volume of the solid obtained by rotating the portion of the region bounded by and that lies in the first quadrant about the y-axis. rewrite the functions: r Volume = Problem courtesy of: Done

Next

2 Volume = x f (x) A bit more theory and Next

Determine the volume of the solid obtained by rotating the region bounded by and the x-axis about the y-axis. Problem courtesy of: Next

2 Volume = x f (x) Determine the volume of the solid obtained by rotating the region bounded by and the x-axis about the y-axis. Problem courtesy of: Done

Questions Back Try the following questions. 1 st Click will bring up a hint. 2 nd click will bring up the answer. 3 rd click will bring up the next question.

Determine the volume of the solid obtained by rotating the region bounded by y = x^3 and y = x, about the x -axis.

Determine the volume of the solid obtained by rotating the region bounded by x = 2, x = 0, and y = 9 about the y -axis. y

Determine the volume of the solid obtained by rotating the region bounded by y = x and, about y = 1.

Determine the volume of the solid obtained by rotating the region bounded by y = 2(x^2) - x^3 and the x -axis, about y -axis. Done