Learning Materials
Features
Discover
Chapter 5: Q45P (page 248)
Find the mass of the solid in Problem 43 if the density is proportional to x.
The required Mass of the solid is .
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Use Problems 12 and 13 to find the centroids of a semi-circular area and of a semi-circular arc. Hint: Assume the formulas , for a sphere.
In the integral
.
Make the change of variables
And evaluate I. Hint: Use (4.8) and the accompanying discussion.
(a) Revolve the curve , from , about the x axis to create a surface and a volume. Write integrals for the surface area and the volume. Find the volume, and show that the surface area is infinite. Hint: The surface area integral is not easy to evaluate, but you can easily show that it is greater than which you can evaluate.
(b) The following question is a challenge to your ability to fit together your mathematical calculations and physical facts: In (a) you found a finite volume and an infinite area. Suppose you fill the finite volume with a finite amount of paint and then pour off the excess leaving what sticks to the surface. Apparently, you have painted an infinite area with a finite amount of paint! What is wrong? (Compare Problem 15.31c of Chapter 1.)
Prove the following two theorems of Pappus: Use Problems 12 and 13 to find the volume and surface area of a torus (doughnut).
Find the volume between the planes z = 2x + 3y +6 and z = 2x + 7y + 8, , and over the square in the (x,y) plane with vertices (0,0) , (1,0) (0,1) (1,1) .
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Exams 
GCSE 
IGCSE 
AS 
A Level 
International A Level 
University Admissions Tests 
Flashcards 
Vaia AI 
Notes 
Study Plans 
Study Sets 
Find a job 
Find a degree 
Student Deals 
Magazine 
Mobile App