Double integral of volume bounded by plane and paraboloid

In summary, the conversation is about evaluating the volume of a solid bounded by the plane z=x and the paraboloid z = x^2 + y^2. The person has graphed the equations and is wondering if they are correct, as it appears that they do not intersect to form a solid. The expert explains that they do intersect, and suggests projecting the intersection onto the xy-plane to visualize it better. They also suggest converting to polar coordinates for a simpler integral.
  • #1
braindead101
162
0
Evaluate the volume of the solid bounded by the plane z=x and the paraboloid z = x^2 + y^2


I have tried to graph this, and they don't bound anything? have i graphed it wrong. and is there a way to do these problems where you don't need to draw the graph.
 
Physics news on Phys.org
  • #2
braindead101 said:
Evaluate the volume of the solid bounded by the plane z=x and the paraboloid z = x^2 + y^2


I have tried to graph this, and they don't bound anything? the graph says they intersect at a line, so they don't bound anything... have i graphed it wrong. and is there a way to do these problems where you don't need to draw the graph.
They surely do bound something! If the 3-d graph bothers you assume y= 0. Then your graphs are z= x and z= x^2, which intersect at (0,0) and (0,1).
Letting z= x in z= x^2+ y^2, you get x^2- x+ y^2= x^2- x+ 1/4 + y^2= 1/4 so
(x-1/2)^2+ y^2= 1/4. In the xy-plane, that's a circle with center at (1/2, 0) and tangent to the y-axis. That's the projection of the actual intersection in the xy-plane.
 
Last edited by a moderator:
  • #3
Graphs can be very useful in this type of problem.

z = x is like a ramp that runs the length of the y axis.

What does the parabola equation describe? Can you visualize it?
 
  • #4
This was also posted in the homework forum. I'm merging it into that thread.
 
  • #5
am i able to change to polar coordinates with the shifted circle? or should i keep it in x y form

let me know if this is correct:
[tex]\int^{1/2}_{0}\int^{\sqrt{1/4-(x-1/2)^{2}}}_{-\sqrt{1/4-(x-1/2)^{2}}} x^{2} + y^{2} dy dx[/tex]

I think i want to convert to polar form.. but i don't know what to do with the 1/4 inside the ( ) with the x when converting.
also, is my double integral even correct?
 
  • #6
If it were me, I would convert to polar coordinates. If [itex](x- 1/2)^2+ y^2= 1/4[/itex], then [itex]x^2+ y^2- x+ 1/4= 1/4[/itex] or [itex]x^2+ y^2- x= 0[/itex] so, in polar coordinates [itex]r^2- rcos(\theta)= 0[/itex] and, finally, [itex]r= cos(\theta)[/itex]. Looks nice!
 

1. What is a double integral of volume?

A double integral of volume is a mathematical tool used to calculate the volume of a three-dimensional solid. It involves integrating a function over a two-dimensional region in the x-y plane.

2. How is the volume bounded by a plane and paraboloid calculated?

The volume bounded by a plane and paraboloid is calculated by setting up a double integral, where the inner integral is taken with respect to the x variable and the outer integral is taken with respect to the y variable. The bounds of the integrals are determined by the intersection points of the plane and paraboloid.

3. Can the order of integration be changed in a double integral of volume?

Yes, the order of integration can be changed in a double integral of volume. This is known as changing the order of integration or using Fubini's theorem, and it allows for easier evaluation of the integral in certain cases.

4. What is the significance of the region bounded by the plane and paraboloid in a double integral of volume?

The region bounded by the plane and paraboloid is important in determining the bounds of the double integral. It represents the three-dimensional solid whose volume is being calculated.

5. What are some applications of calculating double integrals of volume?

Double integrals of volume have many applications in fields such as physics, engineering, and economics. They can be used to calculate the mass of an object with varying density, the work done by a force on a moving object, and the expected value of a random variable in a probability distribution, to name a few.

Similar threads

  • Calculus and Beyond Homework Help
Replies
12
Views
810
  • Calculus and Beyond Homework Help
Replies
2
Views
882
Replies
1
Views
774
  • Calculus and Beyond Homework Help
Replies
6
Views
987
  • Calculus and Beyond Homework Help
Replies
1
Views
954
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
3K
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
Back
Top