Download presentation

Presentation is loading. Please wait.

Published byAbigail Franklin Modified over 2 years ago

1
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 of the pyramid and make down the positive direction, then s=h. 0 3 h This correlates with the formula:

2
Method of Slicing: 1 Find a formula for A ( x ). Sketch the solid and a typical cross section. 2 3 Find the limits of integration. 4 Integrate V ( x ) to find volume.

3
Volume formula :: Let S be a solid bounded by two parallel planes perpendicular to the x-axis at x=a and x=b. If, for each x in [a,b], the cross-sectional area of S perpendicular to the x-axis is A(x), then the volume of the solid is provided A(x) is integrable.

4
Volume formula :: Let S be a solid bounded by two parallel planes perpendicular to the y-axis at y=c and y=d. If, for each y in [c,d], the cross-sectional area of S perpendicular to the y-axis is A(y), then the volume of the solid is provided A(y) is integrable.

5
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 shape several ways, but the simplest cross section is a rectangle. If we let h equal the height of the slice then the volume of the slice is: Since the wedge is cut at a 45 o angle: x h 45 o Since

6
x y Even though we started with a cylinder, does not enter the calculation!

7
Cavalieri’s Theorem: Two solids with equal altitudes and identical parallel cross sections have the same volume. Identical Cross Sections

8
Suppose we start with this curve. If we want to build a nose cone in this shape. So we put a piece of wood in a lathe and turn it to a shape to match the curve.

9
How could we find the volume of the cone? One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes. The volume of each flat cylinder (disk) is: In this case: r= the y value of the function thickness = a small change in x = dx

10
The volume of each flat cylinder (disk) is: If we add the volumes, we get:

11
This application of the method of slicing is called the disk method. The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk. If the shape is rotated about the x-axis, then the formula is: A shape rotated about the y-axis would be:

12
Find the volume of the solid generated by revolving the regions about the x-axis. bounded by

13
Find the volume of the solid generated by revolving the regions about the x-axis.bounded by

14
Find the volume of the solid generated by revolving the regions about the y-axis. bounded by

15
Find the volume of the solid generated by revolving the regions about the x-axis.bounded by

16
Find the volume of the solid generated by revolving the regions about the line y = -1.bounded by

17
The region between the curve, and the y -axis is revolved about the y -axis. Find the volume. y x We use a horizontal disk. The thickness is dy. The radius is the x value of the function. volume of disk

18
The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis: The volume can be calculated using the disk method with a horizontal disk.

19
The region bounded by and is revolved about the y-axis. Find the volume. The “disk” now has a hole in it, making it a “washer”. If we use a horizontal slice: The volume of the washer is: outer radius inner radius

20
This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle. The washer method formula is:

21
If the same region is rotated about the line x = 2 : The outer radius is: R The inner radius is: r

Similar presentations

OK

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 7.3 Volumes of rotation by Disks Limerick Nuclear Generating Station,

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 7.3 Volumes of rotation by Disks Limerick Nuclear Generating Station,

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on ectopic pregnancy Ppt on event driven programming wiki Ppt on power sharing in india download Ppt on fibonacci numbers in art Ppt on transportation in plants for class 10 Ppt on synthesis and degradation of purines and pyrimidines of dna Ppt on surface water treatment Download ppt on android Download ppt on square and square roots for class 8 Ppt on range of motion