Sec 6.2: VOLUMES Volume = Area of the base X height.

Slides:

Advertisements

Similar presentations

Volumes by Slicing: Disks and Washers

Advertisements

DO NOW: Find the volume of the solid generated when the

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.

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,

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.

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 5.2 Volumes. DEFINITION OF VOLUME USING VERTICAL SLICES Let S be a solid that lies between x = a and x = b. If the cross-sectional area of S in.

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

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.

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.

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.

7.2: Volumes by Slicing – Day 2 - Washers Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2001 Little Rock Central High School,

7.3 Day One: Volumes by Slicing Find the volume of the pyramid: Consider a horizontal slice through the pyramid. s dh The volume of the slice.

3 3 3 Find the volume of the pyramid: Consider a horizontal slice through the pyramid. s dh The volume of the slice is s 2 dh. If we put zero at the top.

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.

Lesson 6-2b Volumes Using Discs. Ice Breaker Homework Check (Section 6-1) AP Problem 1: A particle moves in a straight line with velocity v(t) = t². How.

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

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,

7.2 Volumes APPLICATIONS OF INTEGRATION In this section, we will learn about: Using integration to find out the volume of a solid.

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

Chapter 6 Applications of Integration 机动目录上页下页返回结束 6.1 Area Between Curves 6.2 Volume 6.3 Volume by Cylindrical Shell 6.5 Average Value of a Function.

Chapter 7 Quiz Calculators allowed. 1. Find the area between the functions y=x 2 and y=x 3 a) 1/3 b) 1/12 c) 7/12 d) 1/4 2. Find the area between the.

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

Chapter 5: Integration and Its Applications

Inner radius cylinder outer radius thickness of slice.

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

Volumes By Cylindrical Shells Objective: To develop another method to find volume without known cross-sections.

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

Volumes Lesson 6.2.

Disks, Washers and Shells Limerick Nuclear Generating Station, Pottstown, Pennsylvania.

Augustin Louis Cauchy 1789 – 1857 Augustin Louis Cauchy 1789 – 1857 Cauchy pioneered the study of analysis, both real and complex, and the theory of permutation.

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.

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:

Ch. 8 – Applications of Definite Integrals 8.3 – Volumes.

Chapter Area between Two Curves 7.2 Volumes by Slicing; Disks and Washers 7.3 Volumes by Cylindrical Shells 7.4 Length of a Plane Curve 7.5 Area.

6.2 - Volumes Roshan. What is Volume? What do we mean by the volume of a solid? How do we know that the volume of a sphere of radius r is 4πr 3 /3 ? How.

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.

Copyright © Cengage Learning. All rights reserved. 5.2 Volumes

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

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.

Volume is understood as length times width times height, or on a graph, x times y times z. When an integral is revolved around an axis, this is the area.

7.2 Volume: The Disk Method (Day 3) (Volume of Solids with known Cross- Sections) Objectives: -Students will find the volume of a solid of revolution using.

The region enclosed by the x-axis and the parabola is revolved about the line x = –1 to generate the shape of a cake. What is the volume of the cake? DO.

VOLUMES BY CYLINDRICAL SHELLS CylinderShellCylinder.

AP Calculus Honors Ms. Olifer

Review: Area betweens two curves

APPLICATIONS OF INTEGRATION

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.

AP Calculus BC April 3-4, 2017 Mrs. Agnew

Solids not generated by Revolution

6.1 Areas Between Curves To find the area:

6.2 Solids of Revolution-Disk Method Warm Up

6.3 – Volumes By Cylindrical Shells

Presentation transcript:

Sec 6.2: VOLUMES Volume = Area of the base X height

Sec 6.2: VOLUMES Volume = Area of the base X height

Sec 6.2: VOLUMES For a solid S that isn’t a cylinder we first “cut” S into pieces and approximate each piece by a cylinder.

Sec 6.2: VOLUMES

If the cross-section is a disk, we find the radius of the disk (in terms of x ) and use 1 Disk cross-section x animation

Sec 6.2: VOLUMES 1 Disk cross-section x use your imagination

Sec 6.2: VOLUMES 1 Disk cross-section x use your imagination

Sec 6.2: VOLUMES 1 Disk cross-section x step1 Graph and Identify the region step2 Draw a line perpendicular to the rotating line at the point x step3 Find the radius r of the disk in terms of x step4 Now the cross section Area is step5 Specify the values of x step6 The volume is given by

Sec 6.2: VOLUMES

If the cross-section is a washer,we find the inner radius and outer radius Sec 6.2: VOLUMES 2 washer cross-section x

Sec 6.2: VOLUMES step1 Graph and Identify the region step2 Draw a line perpendicular to the rotating line at the point x step3 Find the radius r(out) r(in) of the washer in terms of x step4 Now the cross section Area is step5 Specify the values of x step6 The volume is given by 2 washer cross-section x

Sec 6.2: VOLUMES T-102

Example: Sec 6.2: VOLUMES Find the volume of the solid obtained by rotating the region enclosed by the curves y=x and y=x^2 about the line y=2. Find the volume of the resulting solid.

Sec 6.2: VOLUMES If the cross-section is a disk, we find the radius of the disk (in terms of y ) and use 3 Disk cross-section y

Sec 6.2: VOLUMES 3 Disk cross-section y step1 Graph and Identify the region step2 Draw a line perpendicular to the rotating line at the point y step3 Find the radius r of the disk in terms of y step4 Now the cross section Area is step5 Specify the values of y step6 The volume is given by step2 Rewrite all curves as x = in terms of y

If the cross-section is a washer,we find the inner radius and outer radius Sec 6.2: VOLUMES 4 washer cross-section y Example: The region enclosed by the curves y=x and y=x^2 is rotated about the line x=-1. Find the volume of the resulting solid.

Sec 6.2: VOLUMES 4 washer cross-section y step1 Graph and Identify the region step2 Draw a line perpendicular to the rotating line at the point y step3 Find the radius r(out) and r(in) of the washer in terms of y step4 Now the cross section Area is step5 Specify the values of y step6 The volume is given by step2 Rewrite all curves as x = in terms of y

Sec 6.2: VOLUMES T-091

Sec 6.2: VOLUMES T-072

If the cross-section is a washer,we find the inner radius and outer radius Sec 6.2: VOLUMES 4 washer cross-section y T-102

solids of revolution Sec 6.2: VOLUMES SUMMARY: The solids in all previous examples are all called solids of revolution because they are obtained by revolving a region about a line. Rotated by a line parallel to x-axis ( y=c) Rotated by a line parallel to y-axis ( x=c) NOTE: The cross section is perpendicular to the rotating line solids of revolution Cross-section is DISK Cross—section is WASHER

Sec 6.2: VOLUMES not solids of revolution The solids in all previous examples are all called solids of revolution because they are obtained by revolving a region about a line. We now consider the volumes of solids that are not solids of revolution.

T-102 Sec 6.2: VOLUMES

T-092 Sec 6.2: VOLUMES