Integral Calc: Volume of Solid of Revolution

In summary: The "stick" is running from y=1 to x=0 so that is included in the answer. The "stick" never actually touches the y-y^3 part, so that is included in the answer.
  • #1
cathy
90
0

Homework Statement



Find the volume of the first quadrant region bounded by x=y-y3, x=1 and y=1 that is revolved about the y-axis.



2. The attempt at a solution

v=∏ ∫ from 0 to 1 of (y-y^3)^2 dy
and doing this, I got the answer to be 8∏/105.

Did I set up that integral correctly? I am confused as to if I did this right or not. Please advise. Thank you in advance.
 
Physics news on Phys.org
  • #2
What is y3?
 
  • #3
the equation is x=y-y^3
 
  • #4
cathy said:

Homework Statement



Find the volume of the first quadrant region bounded by x=y-y3, x=1 and y=1 that is revolved about the y-axis.
2. The attempt at a solution

v=∏ ∫ from 0 to 1 of (y-y^3)^2 dy
and doing this, I got the answer to be 8∏/105.

Did I set up that integral correctly? I am confused as to if I did this right or not. Please advise. Thank you in advance.
No, that is not set up correctly. [itex]y- y^3[/itex] is the distance from the y-axis to [itex]x= y- y^3[/itex] but you said the region to be rotated was bounded by [itex]x= y- y^3[/itex] and x=1.

What you have will give the volume from rotating the region bounded by x= 0 and [itex]x= y- y^3[/itex], the region inside the graph while you want the region outside that graph and inside x= 1.

You could use the "washer method" but equivalent, and simpler in my opinion, is to find the volume inside the rotated graph, using the integral as you have it set up, and then subtract that from the volume of the cylinder of radius 1 and height 1 leaving only the volume outside the graph.
 
  • Like
Likes 1 person
  • #5
Okay, I understand what I was doing wrong. I was looking for area inside, rather than outside and within bounds. Thank you.

But, the teacher gave us the answer as V=2∏∫ from 0 to 1 of (y)(1-y+y^3) dy.

And I have a question about this. She said that r=y and h= 1-x, but if you were to draw this as shells, the "stick" would run from the line y=1 to the line x=0. The "stick" never actually touches the y-y^3 part, so how is that included? We take a vertical stick, correct? I'm confused. :(
 
Last edited:
  • #6
And just to clarify, the "r" is talking about the radius of the shell formed, correct?
And since it is a vertical cylinder, why isn't the answer in terms of x?
 
Last edited:

What is the concept of "Integral Calc: Volume of Solid of Revolution"?

The concept of "Integral Calc: Volume of Solid of Revolution" is a mathematical technique used to calculate the volume of a 3-dimensional shape created by rotating a 2-dimensional curve around a fixed axis. This technique involves breaking down the shape into infinitesimally small sections and using integrals to sum up the volumes of these sections.

What are the steps involved in calculating the volume of a solid of revolution?

The steps involved in calculating the volume of a solid of revolution are as follows:
1. Determine the axis of rotation and the bounds of integration.
2. Set up an integral expression that represents the volume of one infinitesimal section of the solid.
3. Integrate the expression over the given bounds to find the total volume of the solid.

What are the different methods used in "Integral Calc: Volume of Solid of Revolution"?

There are two main methods used in "Integral Calc: Volume of Solid of Revolution" - the Disk Method and the Shell Method. The Disk Method involves summing up the volumes of infinitesimal circular disks stacked on top of each other, while the Shell Method involves summing up the volumes of infinitesimal cylindrical shells.

What are the common types of 2-dimensional curves used in "Integral Calc: Volume of Solid of Revolution"?

The common types of 2-dimensional curves used in "Integral Calc: Volume of Solid of Revolution" are:
1. Functions with positive values on the interval of integration.
2. Functions that are bounded above and below by other functions.
3. Functions that cross the x-axis and have negative values on the interval of integration.

What are some real-world applications of "Integral Calc: Volume of Solid of Revolution"?

"Integral Calc: Volume of Solid of Revolution" has many real-world applications, including:
1. Calculating the volume of a water tank or a swimming pool.
2. Finding the volume of a 3-dimensional object created by rotating a 2-dimensional design or logo.
3. Determining the amount of material needed to create a cylindrical or conical structure, such as a silo or a chimney.

Similar threads

Volume of a solid of revolution around the y-axis (def. integration)
    • Calculus and Beyond Homework Help
Replies
3
Views
957
Volume of revolution, region bounded by two functions
    • Calculus and Beyond Homework Help
Replies
1
Views
912
Solid generated by revolving region, find the diameter of the hole
    • Calculus and Beyond Homework Help
Replies
1
Views
909
Calculating the volume of the solid in this graph
    • Calculus and Beyond Homework Help
Replies
4
Views
944
Find the volume of the solid bound by the three coordinate planes
    • Calculus and Beyond Homework Help
Replies
2
Views
449
How to find the volume when solids intersect?
    • Calculus and Beyond Homework Help
Replies
8
Views
3K
Finding the Centre of Mass of a Hemisphere
    • Calculus and Beyond Homework Help
Replies
9
Views
970
Volume of revolution in the first quadrant?
    • Calculus and Beyond Homework Help
Replies
3
Views
2K
Volume of solid region double integral
    • Calculus and Beyond Homework Help
Replies
27
Views
2K
Volume of a solid of rotation, obtained rotating a function around x=2
    • Calculus and Beyond Homework Help
Replies
2
Views
868

Hot Threads

Recent Insights

Back
Top