We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you!
Presentation is loading. Please wait.
Published byLindsey Budd
Modified over 7 years ago
Applications of Integration Volumes of Revolution Many thanks to http://mathdemos.gcsu.edu/shellmeth od/gallery/gallery.html
Method of discs
Take this ordinary line 2 5 Revolve this line around the x axis We form a cylinder of volume
We could find the volume by finding the volume of small disc sections 2 5
If we stack all these slices… We can sum all the volumes to get the total volume
To find the volume of a cucumber… we could slice the cucumber into discs and find the volume of each disc.
The volume of one section: Volume of one slice =
We could model the cucumber with a mathematical curve and revolve this curve around the x axis… Each slice would have a thickness dx and height y. 25 -5
The volume of one section: r = y value h = dx Volume of one slice =
Volume of cucumber… Area of 1 slice Thickness of slice
Take this function… and revolve it around the x axis
We can slice it up, find the volume of each disc and sum the discs to find the volume….. Radius = y Area = Thickness of slice = dx Volume of one slice=
Take this shape…
Divide the region into strips
Form a cylindrical slice
Repeat the procedure for each strip
To generate this solid
Regions that can be revolved using disc method
Regions that cannot….
Model this muffin.
A different cake
Making a washer
Revolving around the x axis
Region bounded between y = 1, x = 0, y = 1 x = 0
Volume generated between two curves y= 1
Area of cross section.. f(x) g(x)
Your turn: Region bounded between x = 0, y = x,
Region bounded between y =1, x = 1
Region bounded between
Around the x axis- set it up
Revolving shapes around the y axis
Region bounded between
Volume of one washer is
Calculate the volume of one washer
And again…region bounded between y=sin(x), y = 0.
Region bounded between x = 0, y = 0, x = 1,
Worksheet 5 Delta Exercise 16.5
Volumes by Slicing: Disks and Washers
Disk and Washer Methods
DO NOW: Find the volume of the solid generated when the
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.
A k = area of k th rectangle, f(c k ) – g(c k ) = height, x k = width. 6.1 Area between two curves.
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.
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.
The Shell Method Volumes by Cylindrical Shells By Christine Li, Per. 4.
Volume: The Disk Method
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.
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.
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.
7.3 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.
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.
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.
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.
© 2022 SlidePlayer.com Inc. All rights reserved.