Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

Find the volume common to two circular cylinders,…

01:18

Question

Answered step-by-step

Problem 65 Hard Difficulty

(a) Cavalieri's Principle states that if a family of parallel planes gives equal cross-section areas for two solids $ S_1 $ and $ S_2 $ then the volumes of $ S_1 $ and $ S_2 $ are equal. Prove this principle.
(b) Use Cavalieri's Principle to find the volume of the oblique cylinder shown in the figure.


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Carson Merrill
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Carson Merrill

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

More Answers

01:58

WZ

Wen Zheng

Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 6

Applications of Integration

Section 2

Volumes

Related Topics

Applications of Integration

Discussion

You must be signed in to discuss.
Top Calculus 2 / BC Educators
Catherine Ross

Missouri State University

Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

Samuel Hannah

University of Nottingham

Calculus 2 / BC Courses

Lectures

Join Course
Recommended Videos

03:43

(a) Cavalieri's Princ…

03:22

(a) Cavalieri's Princ…

02:00

Cavalieri's principle…

00:24

In 1635 Bonaventura Cavali…

Watch More Solved Questions in Chapter 6

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Problem 63
Problem 64
Problem 65
Problem 66
Problem 67
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72

Video Transcript

So for this problem, we're gonna let A s one and A s to represent the areas of the cross sections of the solids s one and s two. So the volume of s one will be equal to are bound the integral of our bounds a one to be one of a s one DX and similarly, we have a volume of the S two, which is a 22 b two a s, two d x some. Based on this, we're assuming that s one equals s two. Because the planes of the cross sections are parallel, we know that the bounds of the integral czar equal so a one equals a two and B one equals b two. So based on this, since these two areas are equal to each other and then we also have that the bounds are equal to each other. We can conclude that a one B one A s one D x is equal to a to B two a s, two DX Um So what that ultimately means is that V S one is equal to V S two than for Part B. Um, we have a given cylinder C one And then we have another cylinder, C two with height each and radius arm, and we can assume that they're lower bases are in the same plane. Um, using the principle that we just discussed. What we have is that the volume of the first one equals the Honourable from zero to h of a one x dx, and that's going to be equal r squared pi times in a row from zero to h dx. And that's just r squared h Then we also do is V two, and we end up getting that. This is going to give us the same result of R squared pi H, and we already knew that their bounds were equal. So we see that since their volumes are equal, um, we can set up those cylinders on the same plane, and it's clear that they have the same intersection with corresponding plane. Um, and we see that their volumes are equal, which is what we wanted to show. And this is R squared pi H, by the way,

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
151
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
68
Hosted by: Alonso M
See More

Related Topics

Applications of Integration

Top Calculus 2 / BC Educators
Catherine Ross

Missouri State University

Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

Samuel Hannah

University of Nottingham

Recommended Videos

03:43

(a) Cavalieri's Principle states that if a family of parallel planes gives equa…

03:22

(a) Cavalieri's Principle states that if a family of parallel planes gives equa…

02:00

Cavalieri's principle states that if two solids of equal altitudes have the sam…

00:24

In 1635 Bonaventura Cavalieri, a student of Galileo, stated the following resul…
Additional Mathematics Questions

01:14

nathan has 4 brown teddy bears, 5 pink teddy bears and 3 red teddy bears in …

00:52

What is the slope of the line through
(
−
7
,
−
2
)
…

01:03

A football player scores 40 goals in 60 games. find his rate of scoring per …

00:39

Mina bought a plane ticka to New York City and used a coupon for 10% off the…

01:41

Kerrie uses the work shown below to determine 35% of 280.
Step 1: 35% = 1…

05:15

The product of two consecutive positive integers is 1332 explain how you can…

01:14

A parallelogram has a base of 4.5 cm and an area of 9.495 cm². Tania wrote t…

01:18

In the last ten games, Jamal made 7
9
of his free throws and Brian ma…

01:18

Britney is solving a quadratic equation. Her first
step is shown below.

00:47

A charity organization is holding a food drive with a goal to collect at lea…

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started