Visualising solid for double integration.

In summary, ProPatto16 struggles to visualize the solid and is not sure how to illustrate it. Someone suggested that he draw the xy plane and show the base of the shape. This gives him the information he needs to figure out the volume of the solid.
  • #1
ProPatto16
326
0

Homework Statement



find volume of solid bounded by planes z=x, y=x, x+y=2, z=0

im struggling to visualise the solid.

can anyone help me? I am not sure how so illustrate but any pointers would be appreciated!

thank you!
 
Physics news on Phys.org
  • #2
Hi ProPatto16! :smile:
ProPatto16 said:
z=x, y=x, x+y=2, z=0

Start with the easy ones … y=x, x+y=2.

These are both vertical planes, at a right-angle to each other: one intersects the "ground" at the 45° line through the origin, the other at the minus 45° line, through … ?

Then z = x is a slope through the y axis, and of course z = 0 is the "ground". :wink:
 
  • #3
Yeah I actually went and got an ice cream container and squares of card and tried to make it haha. With the x-axis along the bottom and the y up the left side, I worked out all the planes but then there seems to be no top on the shape. And I need another vertical plane to bound the shape along the x axis?

I'm sorry this is so ambiguous. Once I get the shape I know I can do the question.
 
  • #4
Also I am advised to draw the xy plane showing the base of the shape, ie the area I'm integrating over. So in this case it would be the xy plane with lines y=x and y=2-x which gives a cross when x and y-axis go from 0-2 and the two lines intersect at 1,1. So which of those 4 triangles in that xy plane is the one I'm integrating over? The one on the left bounded by y? Or the one on the bottom bounded by x? The top one and right one can be omitted since not bounded by both curves.
 
  • #5
ProPatto16 said:
… I worked out all the planes but then there seems to be no top on the shape.

It has a sloping top, x = z :wink:
And I need another vertical plane to bound the shape along the x axis?
… So which of those 4 triangles in that xy plane is the one I'm integrating over? The one on the left bounded by y? Or the one on the bottom bounded by x? The top one and right one can be omitted since not bounded by both curves.

hmm :rolleyes: …

you're right, aren't you? :smile:

eg every point (1,y,0.5) will be in the region, for any y < 1. :biggrin:

(i expect they meant to say "y = z = 0" :redface: … try it like that! :rolleyes:)
 
  • #6
Okay I think I see it now. My piece of card for x=z was down the side of my shape lol!
Ill see how I go and may reply back again tomorrow.
thanks! :)
 
  • #7
okay so I've come to this. since z = x

take double integral of x.dydx with x<y<2-x and 0<x<1

but since I am integrating by y first, and there is no y, its just the integral of x.dx between x and 2-x which becomes the integral of (2-2x).dx between 0 and 1.

which is 2x-x2 between 0 and 1... subbing in 1 gives 2-1=1

so the volume of the solid is 1 unit cubed?
 

FAQ: Visualising solid for double integration.

What is the double integration method used for?

The double integration method is used in mathematics and physics to calculate the volume of a solid in three-dimensional space. It involves integrating a function twice, first with respect to one variable and then with respect to another variable.

How is a solid visualized for double integration?

A solid for double integration is visualized as a three-dimensional object, with each point in the solid having a specific value of the function being integrated. The boundaries of the solid can be defined by equations or inequalities for the two variables being integrated.

What is the difference between single and double integration?

Single integration involves calculating the area under a curve in two-dimensional space, while double integration calculates the volume of a solid in three-dimensional space. Double integration involves integrating a function twice, while single integration involves only one integration.

How is the order of integration determined?

The order of integration is determined by the orientation of the solid and the chosen variables. The inner integral is calculated first, with respect to the variable that changes more frequently. The outer integral is then calculated with respect to the remaining variable.

What are some real-world applications of double integration?

Double integration has many real-world applications, including calculating the area under a curved roof, determining the volume of a 3D-printed object, and finding the mass of an irregularly shaped object. It is also used in physics to calculate the work done by a force on an object.

Similar threads

Integrating Multiple Solid Volumes: xy-Plane, x=1, x=3
Replies
1
Views
1K
Calculating the volume of the solid in this graph
Replies
4
Views
1K
Finding Volume of Solid Bounded by Paraboloid and Planes
Replies
2
Views
2K
How to determine the volume of a region bounded by planes?
Replies
4
Views
1K
Cylindrical Coordinates Triple Integral -- stuck in one place
Replies
1
Views
2K
Volume of a solid of revolution around the y-axis (def. integration)
Replies
3
Views
1K
Double integral, find volume of solid
Replies
2
Views
5K
Triple Integral To Find Volume Between Cylinder And Sphere
Replies
2
Views
334
Solid generated by revolving region, find the diameter of the hole
Replies
1
Views
1K
Where Can I Find Resources to Help Me Solve Double Integral Problems?
Replies
9
Views
1K

Hot Threads

Recent Insights

Back
Top