How do I calculate the volume of this?

  • Thread starter Thread starter -EquinoX-
  • Start date Start date
  • Tags Tags
    Volume
Click For Summary
SUMMARY

The volume of the region bounded by the curves y = x^2, y = 1, and the y-axis, when rotated around the y-axis, can be calculated using the disk method. The volume function V(y) is defined such that dV/dy = πy, where y represents the radius squared of the circular cross-section. The correct integral to evaluate this volume is from 0 to 1 of πy dy, leading to the formula V = ∫(0 to 1) πy dy. The confusion regarding vertical versus horizontal slicing was clarified, emphasizing that horizontal slices yield the correct circular disks for this rotation.

PREREQUISITES
  • Understanding of calculus concepts, specifically integration
  • Familiarity with the disk method for calculating volumes of revolution
  • Knowledge of the equations of curves, particularly parabolas and lines
  • Ability to manipulate and evaluate definite integrals
NEXT STEPS
  • Study the disk method in detail, focusing on volume calculations for solids of revolution
  • Learn about the washer method for cases where the region has inner and outer radii
  • Explore applications of integration in calculating areas and volumes in multivariable calculus
  • Practice solving volume problems involving different curves and axes of rotation
USEFUL FOR

Students and educators in calculus, particularly those studying volume calculations through integration, as well as anyone interested in solid geometry and applications of the disk method.

-EquinoX-
Messages
561
Reaction score
1
How do I calculate the volume of this??

What is the volume of the region bounded by y = x^2, y = 1, and the y-axis rotated around the y -axis
 
Physics news on Phys.org
define a voklume function, as a function of y, by letting V(y) be that portion of the volume lying below height y. then the derivative of this function i the area of the circular face of this portion of volume, i.e. dV/dy = πr^2 where r = x = sqrt(y), so dV/dy = πy. so you guess a formula for V(y) and then plug in y = 1.
 
I should do this using integral.. I was thinking of slicing this region vertically but then how do I represent this in integrals?? I have to take integrals from -1 to 1 right?

I would evaluate this as the integral from 0 to 1 of (1-squareroot of y)^2 dy

is that right??
 
Last edited:
Why in the world would you slice it vertically? Since it is rotated around the y-axis, slices horizontally will be disks with radius x. Each would have area \pi x^2= \pi y and each infinitesmal disc will have volume \pi y dy. Integrate that.
 
HallsofIvy said:
Why in the world would you slice it vertically? Since it is rotated around the y-axis, slices horizontally will be disks with radius x. Each would have area \pi x^2= \pi y and each infinitesmal disc will have volume \pi y dy. Integrate that.


I am sorry that's my mistake. I would slice it horizontally and take the integral from 0 to 1. And shouldn't the radius be 1-square root of y?? Because it's the intersection of y = x^2 and y = x
 
-EquinoX- said:
I am sorry that's my mistake. I would slice it horizontally and take the integral from 0 to 1. And shouldn't the radius be 1-square root of y?? Because it's the intersection of y = x^2 and y = x
Where did you get y= x from? Your original post was:

-EquinoX- said:
What is the volume of the region bounded by y = x^2, y = 1, and the y-axis rotated around the y -axis
Every horizontal "slice" is a circle with center at the y-axis, x= 0, and the end of a radius at x= \sqrt{y}. Of course, the area of the disk is \pi x^2= \pi y.
 

Thread 'How does this derivative work? And why the speed of light remains?'

Relativistic Momentum, Mass, and Energy Momentum and mass (...), the classic equations for conserving momentum and energy are not adequate for the analysis of high-speed collisions. (...) The momentum of a particle moving with velocity ##v## is given by $$p=\cfrac{mv}{\sqrt{1-(v^2/c^2)}}\qquad{R-10}$$ ENERGY In relativistic mechanics, as in classic mechanics, the net force on a particle is equal to the time rate of change of the momentum of the particle. Considering one-dimensional...
View full post »

Similar threads

MHB What is the Volume of a Solid with Squares as Cross Sections?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Question about Integrals to Determine Volume vs Surface Area
  • · Replies 5 ·
Replies
5
Views
3K
MHB 18 Find the volume of the solid generated by revolving the region about y-axis.
  • · Replies 4 ·
Replies
4
Views
2K
MHB Triple Integrals: Finding Volume of Solid S Bounded by Planes
  • · Replies 5 ·
Replies
5
Views
2K
High School What Methods Calculate the Volume of a Solid Revolved Around Non-Standard Axes?
  • · Replies 2 ·
Replies
2
Views
2K
High School Question about volume of a sphere
  • · Replies 16 ·
Replies
16
Views
3K
Undergrad Problem with calculating projections of curl using rotation of contour
  • · Replies 9 ·
Replies
9
Views
984
MHB Calculate volume of a solid rotating around the y-axis
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Double integrals - do areas cancel?
  • · Replies 24 ·
Replies
24
Views
4K
MHB Decide volume given two functions
  • · Replies 3 ·
Replies
3
Views
1K