Sketching and Calculating the Volume of Solid E

In summary, the conversation discusses sketching the solid E bounded by a cylinder and two planes, and finding its volume using a triple integral. There is confusion about the planes and their intersection points, and the need for an upper limit on x for the solid to be enclosed.
  • #1
kieranl
24
0

Homework Statement


Sketch the solid E bounded by the cylinder x = y^2 and the planes z = 3 and x + z = 1, and write down its analytic expression. Then, use a triple integral to find the volume of E.

The Attempt at a Solution


Was wondering if someone could have a go at drawing this sketch? In mine, I thought i did it right but can't seem to obtain an enclosed surface. If x+z=1 was rather x-z=1 i would be able to but can't so far?
 
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  • #2
The plane x+ z= 1 crosses the plane z= 3 when x+ 3= 1 or the line x= -2, z= 3, y= t. It crosses the cylinder [itex]x= y^2[/itex] in the line [itex]x= t^2[/itex], [itex]y= t[/itex], [itex]z= 1- x= 1- t^2[/itex].

I wonder if you weren't confusing [itex]x= y^2[/itex] with the [itex]y= x^3[/itex].
 
  • #3
I don't think you NEED to graph this.

y=(plus/minus) sqrt(x)

If that helps.

The range is x>0, so sqrt(x) is real.

If x>0, which is on top, z=3 or z=1-x?EDIt: Is there an upper bound on x?
 
  • #4
wat i have done is drawn the cylinder x = y^2 in the x-y plane and extended it along the z plane. Then i drew the z=3 plane and then drew a line z=1-x and it extended it along the y plane. But this does not end up enclosing a solid
 
  • #5
kieranl said:
wat i have done is drawn the cylinder x = y^2 in the x-y plane and extended it along the z plane. Then i drew the z=3 plane and then drew a line z=1-x and it extended it along the y plane. But this does not end up enclosing a solid

You are right, I think. There must be an upper x-limit for this to be a solid. Maybe, you have accidentally skipped some information.
 

FAQ: Sketching and Calculating the Volume of Solid E

1. What is sketching and calculating the volume of solid E?

Sketching and calculating the volume of solid E is a mathematical process used to determine the amount of space occupied by a three-dimensional shape called solid E. It involves using geometric principles and formulas to find the volume of the shape.

2. Why is it important to sketch and calculate the volume of solid E?

Sketching and calculating the volume of solid E is important because it allows us to accurately measure and compare the sizes of three-dimensional objects. It is also a crucial step in many real-world applications, such as architecture and engineering, where precise measurements are necessary.

3. What are the steps involved in sketching and calculating the volume of solid E?

The steps may vary depending on the specific shape of solid E, but generally, they include identifying the dimensions of the shape, applying the appropriate formula for finding volume, and performing the necessary calculations. This may also involve sketching the shape and labeling the dimensions to help visualize the problem.

4. What are some common formulas used to calculate the volume of solid E?

The formula used to calculate the volume of solid E will depend on its shape. For example, the formula for a cube is V = s^3, where s is the length of one side. For a cylinder, the formula is V = πr^2h, where r is the radius of the base and h is the height. Other common formulas include those for rectangular prisms, pyramids, and spheres.

5. Can sketching and calculating the volume of solid E be used for irregular shapes?

Yes, sketching and calculating the volume of solid E can be used for irregular shapes. It may be more challenging as it may involve breaking the shape down into smaller, simpler shapes and using the appropriate formulas for each. However, with careful measurements and calculations, the volume of irregular shapes can also be determined accurately.

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