Download presentation

Presentation is loading. Please wait.

Published byAdolfo Raffield Modified over 4 years ago

1
Volumes by Disks and Washers Or, how much toilet paper fits on one of those huge rolls, anyway??

2
A Real Life Situation Wow, thats a lot of toilet paper! I wonder how much is actually on that roll? Relief

3
How do we get the answer? CALCULUS!!!!! (More specifically: Volumes by Integrals)

4
Volume by Slicing Volume = length x width x height Total volume = (A x th) Volume of a slice = Area of a slice x Thickness of a slice A th

5
Volume by Slicing Total volume = (A x th) VOLUME = A d(th) But as we let the slices get infinitely thin, Volume = lim (A x th) t 0 Recall: A = area of a slice

6
x=f(y) Rotating a Function Such a rotation traces out a solid shape (in this case, we get a paraboloid) x=f(y)

7
Volume by Slices Slice r } dy Thus, the area of a slice is r^2 A = r^2

8
Disk Formula VOLUME = A d(th) VOLUME = r^2 d(th) But: A = r^2, so… The Disk Formula

9
Volume by Disks r } thickness x axis y axisSlice radius x x dy Thus, A = x^2 x = f(y) VOLUME = f(y)^2 dy but x = f(y)and d(th) = dy, so...

10
More Volumes f(x) g(x) rotate around x axis Slice R r Area of a slice = (R^2-r^2) dx

11
Washer Formula VOLUME = A d(th) VOL = (R^2 - r^2) d(th) But: A = (R^2 - r^2), so… The Washer Formula

12
Volumes by Washers f(x) g(x) Slice R r dt Big R little r g(x) f(x) Thus, A = (R^2 - r^2) dx = (f(x)^2 - g(x)^2) V = (f(x)^2 - g(x)^2) dx

13
2 The application weve been waiting for... 1 rotate around x axis 1 0.5 f(x) g(x)

14
Toilet Paper f(x) g(x) 1 2 0.5 1 So we see that: f(x) = 2, g(x) = 0.5 0 V = (f(x)^2 - g(x)^2) dx x only goes from 0 to 1, so we use these as the limits of integration. Now, plugging in our values for f and g: V = (2^2 - (0.5)^2) dx = 3.75 (1 - 0) = 3.75 0 1

15
Other Applications? Just how much pasta can Pavarotti fit in that tummy of his?? Feed me!!!!!! or,If youre a Britney fan, like say...

16
"Me 'n Britney 4 eva. (I know Mrs. Harlow didnt do this!)

17
Britney You can figure out just how much air that head of hers can hold! Approximate the shape of her head with a function,

18
The Recipe n and Integrate n Slice n Rotate

19
And some people might say that calculus is boring... On the next episode of math fun... Volumes by Shells (aka TP Method)

Similar presentations

OK

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

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

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on four factors of production Ppt on travel and tourism industry Ppt on shell scripting and logic Liquid crystal on silicon display ppt on tv Resource based view ppt online Ppt on mars one hoax Ppt on travelling salesman problem using genetic algorithm Good health habits for kids ppt on batteries Ppt on phonetic transcription vowels Ppt on biodegradable and non biodegradable materials landfill