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.

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 of a Solid The volumes of solid that can be cut into thin slices, where the volumes can be interpreted as a definite integral.

Section Volumes by Slicing

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.

Solids of Revolution Washer Method

Volumes – The Disk Method Lesson 7.2. Revolving a Function Consider a function f(x) on the interval [a, b] Now consider revolving that segment of curve.

7.1 Areas Between Curves To find the area: divide the area into n strips of equal width approximate the ith strip by a rectangle with base Δx and height.

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.

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

7.1 Area Between 2 Curves Objective: To calculate the area between 2 curves. Type 1: The top to bottom curve does not change. a b f(x) g(x) *Vertical.

Review Problem – Riemann Sums Use a right Riemann Sum with 3 subintervals to approximate the definite integral:

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.

 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.

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.

V OLUMES OF SOLIDS WITH KNOWN CROSS SECTIONS 4-H.

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.

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, B Volumes by the Washer Method Limerick Nuclear Generating Station,

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.

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.

Chapter 5: Integration and Its Applications

Solids of Revolution Disk Method

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

Finding Volumes Chapter 6.2 February 22, In General: Vertical Cut:Horizontal Cut:

Volumes Lesson 6.2.

Section 7.3 Volume: The Shell Method. When finding volumes of solids by the disk (or washer) method we were routinely imagining our ‘slices’ under the.

Finding Volumes. In General: Vertical Cut:Horizontal Cut:

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.

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.

7.2 Volume: The Disc Method The area under a curve is the summation of an infinite number of rectangles. If we take this rectangle and revolve it about.

Volume: The Shell Method 7.3 Copyright © Cengage Learning. All rights reserved.

5.2 Volumes of Revolution: Disk and Washer Methods 1 We learned how to find the area under a curve. Now, given a curve, we form a 3-dimensional object:

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.

6.3 Volumes of Revolution Fri Feb 26 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.

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

Volumes of Solids of Rotation: The Disc Method

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

VOLUMES BY CYLINDRICAL SHELLS CylinderShellCylinder.

Solids of Revolution Revolution about x-axis. What is a Solid of Revolution? Consider the area under the graph of from x = 0 to x = 2.

Solids of Revolution Shell Method

Solids of Revolution Shell Method

7.2 Volume: The Disk Method

Finding Volumes Disk/Washer/Shell

Review of Area: Measuring a length.

Finding Volumes Disk/Washer/Shell

Volumes of Solids of Revolution

Chapter 7.2: Volume The Disk Method The Washer Method Cross-sections

9.4: Finding Volume using Washer Method

7.3 Volume: The Shell Method

6.2 Volumes If a region in the plane is revolved about a line, the resulting solid is called a solid of revolution, the line is called the axis of revolution.

Integration Volumes of revolution.

AP Calculus AB Day 3 Section 7.2 5/7/2019 Perkins.

6.2 Solids of Revolution-Disk Method Warm Up

Solids of Revolution PART DEUX: WaShErS Rita Korsunsky

Presentation transcript:

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 in terms of x Revolve about a vertical axis Slice perpendicular to axis – slices horizontal Integrate in terms of y

Washer Method

Volume of Washer: Horizontal axis of revolution Slice with rectangle perpendicular to axis Integrate in terms of x Vertical axis of revolution Slice with rectangles perpendicular to axis Integrate in terms of y Find Volume – Washer Method

1) Find the volume of the solid is bounded by the curve y = x and the x-axis between x = 0 and x = 1 rotated about the x-axis

2)Find the volume of the solid when the region bounded by is revolved about the x-axis

3) Find the volume of the solid generated when the region bounded by is revolved about the x-axis

4) Find the volume of the solid generated when the region bounded by is revolved about the line y = 4

5) Set up the integrals for finding the volume of the solid formed by revolving the region formed by a) About the x-axisb) about y = -1

6) Find the volume of the solid generated when the region bounded by is revolved about the line y = 3

7) Find the volume of the solid generated when the region bounded by is revolved about the y-axis

H OME W ORK Finding Volumes of solids using washers and disks worksheet