That is the integral that I would use to find the volume of the solid.

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In summary, the integral used to find the volume of the solid described below the plane z+y=1 and inside the cylinder x^2+y^2 equal or less than 1, with 0 equal
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Pixter
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q: (set up the integral that you would use to find the volume of the solid described, evaluate the integral)
The region below the plane z+y=1 and inside the cylinder x^2+y^2 equal or less than: 1,0 equal and less to z equal and less to 1

this is what i have been trying:
tried
-sqrt1-x^2 <= y <= sqrt1-x^2
0 <= z <= 1-y
-1 <= x <= 1

the books gives the answer: pi-2/3

oki guys/gals, really can't do it and it's just f--king enoying me soo much.. so if someone could do this or at least give me a sweet description how to do this.. !spending more time on this than my actual revision!
 
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You should show some work, but it sounds like you're really stuck. Start by drawing a picture. If you look at the part of this region for y < 0, you get half a cylinder. If you want to find the volume of this by integrating along the y-axis, then think of this volume as a bunch of thin sheets side by side of height (in the z-direction) 1, width (in the x-direction) [itex]2\sqrt{1-y^2}[/itex], and thickness (in the y-direction) dy. The other half of this irregular shape is more irregular, because the height changes, but you can apply the same idea, and get an expression for the volume of this shape in terms of integrals. Set up an expression, and try evaluating. If you still get stuck, show your work.
 
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  • #3
Pixter said:
q: (set up the integral that you would use to find the volume of the solid described, evaluate the integral)
The region below the plane z+y=1 and inside the cylinder x^2+y^2 equal or less than: 1,0 equal and less to z equal and less to 1

this is what i have been trying:
tried
-sqrt1-x^2 <= y <= sqrt1-x^2
0 <= z <= 1-y
-1 <= x <= 1

the books gives the answer: pi-2/3

oki guys/gals, really can't do it and it's just f--king enoying me soo much.. so if someone could do this or at least give me a sweet description how to do this.. !spending more time on this than my actual revision!

Looks to me like a good candidate for polar coordinates. In polar coordinate, r runs from 0 to 1 and [itex]\theta[/itex] from 0 to [itex]2\pi[/itex]. For each r and [itex]\theta[/itex], z runs from 0 to [itex]1- y= 1- rsin(\theta)[/itex]. Of course, the "differential of area" in polar coordinates is [itex]r dr d\theta[/itex].

The volume is
[tex]\int_{r= 0}^1 \int_{\theta= 0}^{2\pi} (1- r sin(\theta))rd\theta dr[/tex]
 

1. What is the definition of volume of irregular shape?

The volume of an irregular shape refers to the amount of space that the object occupies. It is a measure of the total three-dimensional space enclosed by the shape.

2. How is the volume of an irregular shape calculated?

The volume of an irregular shape can be calculated by using various mathematical formulas depending on the shape. For example, the volume of a irregular solid can be calculated by dividing the object into smaller regular shapes and then adding their volumes together.

3. Are there any tools or methods to measure the volume of an irregular shape?

Yes, there are various tools and methods that can be used to measure the volume of an irregular shape. One common method is to use a graduated cylinder or beaker and measure the amount of water displaced by the object. There are also specialized tools such as laser scanners and 3D software that can accurately measure the volume of complex irregular shapes.

4. Can the volume of an irregular shape be negative?

No, the volume of an irregular shape cannot be negative as it is a measure of physical space and cannot have a negative value. However, the calculated volume of an irregular shape may be negative if the measurements or calculations are incorrect.

5. How is the volume of an irregular shape useful in real-world applications?

The volume of an irregular shape is useful in various real-world applications such as engineering, architecture, and manufacturing. It is an important factor in determining the strength and stability of structures, as well as in calculating materials needed for construction. It is also used in the design and production of various consumer products such as packaging and containers.

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