Learn about different polar graphs, including limaçons (convex, dimpled, looped), lemniscates, rose curves, and cardioids. View compare and contrast between the 4 different types of polar graphs, and view my impressions on this final unit in pre-calculus honors.
linear transformation and rank nullity theorem Manthan Chavda
In these notes, I will present everything we know so far about linear transformations.
This material comes from sections in the book, and supplemental that
I talk about in class.
In this video we learn how to solve limits that involve trigonometric functions. It is all based on using the fundamental trigonometric limit, which is proved using the squeeze theorem.
For more lessons: http://www.intuitive-calculus.com/solving-limits.html
Watch video: http://www.youtube.com/watch?v=1RqXMJWcRIA
This ppt covers following topic of unit - 1 of B.Sc. 1 Calculus :- Definition of limit , left & right hand limit and its example , continuity & its related example.
this is the ppt on application of integrals, which includes-area between the two curves , volume by slicing , disk method , washer method, and volume by cylindrical shells,.
this is made by dhrumil patel and harshid panchal.
Learn about different polar graphs, including limaçons (convex, dimpled, looped), lemniscates, rose curves, and cardioids. View compare and contrast between the 4 different types of polar graphs, and view my impressions on this final unit in pre-calculus honors.
linear transformation and rank nullity theorem Manthan Chavda
In these notes, I will present everything we know so far about linear transformations.
This material comes from sections in the book, and supplemental that
I talk about in class.
In this video we learn how to solve limits that involve trigonometric functions. It is all based on using the fundamental trigonometric limit, which is proved using the squeeze theorem.
For more lessons: http://www.intuitive-calculus.com/solving-limits.html
Watch video: http://www.youtube.com/watch?v=1RqXMJWcRIA
This ppt covers following topic of unit - 1 of B.Sc. 1 Calculus :- Definition of limit , left & right hand limit and its example , continuity & its related example.
this is the ppt on application of integrals, which includes-area between the two curves , volume by slicing , disk method , washer method, and volume by cylindrical shells,.
this is made by dhrumil patel and harshid panchal.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
2. The Disk Method
If a region in the plane is revolved about a line, the resulting
solid is a solid of revolution, and the line is called the axis
of revolution.
The simplest such solid is a right
circular cylinder or disk, which is
formed by revolving a rectangle
about an axis adjacent to one
side of the rectangle,
as shown in Figure 7.13.
Figure 7.13
2
3. The Disk Method
Divide the region into n equal disks
Find the volume of each disk, to get an approximate
volume.
As the number of disks approaches infinity, we
approach the actual volume:
ANIMATION
Figure 7.13
3
5. The Washer Method
The disk method can be extended to cover solids of
revolution with holes by replacing the representative disk
with a representative washer.
The washer is formed by revolving
a rectangle about an axis,
as shown in Figure 7.18.
If r and R are the inner and outer radii
of the washer and w is the width of the
washer, the volume is given by
Volume of washer = π(R2 – r2)w.
5
Figure 7.18
6. The Washer Method
To see how this concept can be used to find the volume of
a solid of revolution, consider a region bounded by an
outer radius R(x) and an inner radius r(x), as shown in
Figure 7.19.
Figure 7.19
6
7. Example 3 – Using the Washer Method
Find the volume of the solid formed by revolving the region
bounded by the graphs of
about the
x-axis, as shown in Figure 7.20.
Figure 7.20
7
8. The Washer Method
If the region is revolved about its axis of revolution, the
volume of the resulting solid is given by
Note that the integral involving the inner radius represents
the volume of the hole and is subtracted from the integral
involving the outer radius.
8
9. Example 3 – Solution
In Figure 7.20, you can see that the outer and inner radii
are as follows.
Integrating between 0 and 1 produces
9
11. Practice
Find the volume of the solid formed by revolving
the region bounded by the graphs of f(x)= - x^2 +5x+3
and g(x) = -x + 8 about the x-axis.
SETUP the integral and then finish with your calculator.
Approximately 442.34 cubic units
11
12. The Washer Method
So far, the axis of revolution has been horizontal and you
have integrated with respect to x. In the Example 4, the
axis of revolution is vertical and you integrate with respect
to y. In this example, you need two separate integrals to
compute the volume.
12
13. Example 4 – Integrating with Respect to y, Two-Integral Case
Find the volume of the solid formed by revolving the region
bounded by the graphs of y = x2 + 1, y = 0, x = 0, and x = 1
about y-axis, as shown in Figure 7.21.
Figure 7.21
13
14. Example 4 – Solution
For the region shown in Figure 7.21, the outer radius is
simply R = 1.
There is, however, no convenient formula that represents
the inner radius.
When 0 ≤ y ≤ 1, r = 0, but when 1 ≤ y ≤ 2, r is determined
by the equation y = x2 + 1, which implies that
14
15. Example 4 – Solution
cont’d
Using this definition of the inner radius, you can use two
integrals to find the volume.
15
16. Example 4 – Solution
Note that the first integral
cont’d
represents the volume
of a right circular cylinder of radius 1 and height 1.
This portion of the volume could have been determined
without using calculus.
16
18. Solids with Known Cross Sections
With the disk method, you can find the volume of a solid
having a circular cross section whose area is A = πR2.
This method can be generalized to solids of any shape, as
long as you know a formula for the area of an arbitrary
cross section.
Some common cross sections are squares, rectangles,
triangles, semicircles, and trapezoids.
18
20. Example 6 – Triangular Cross Sections
Find the volume of the solid shown in Figure 7.25.
The base of the solid is the region bounded by the lines
and x = 0.
Figure 7.25
The cross sections perpendicular to the x-axis are
equilateral triangles.
20
21. Example 6 – Solution
The base and area of each triangular cross section are as
follows.
21
22. Example 6 – Solution
cont’d
Because x ranges from 0 to 2, the volume of the solid is
22