Solving Solid of Revolution HW: Find V(L) for 0<=L<=2R

In summary, Peter has a spherical water tank with a radius of R and wants to know the water volume at a distance L from the hole at the top of the tank. Solutions were given using integration, but @Kqwert's solution was incorrect and yielded the volume of a sphere instead of the correct solution provided in the solution manual. The correct solution involves adding the two expressions provided in post #3 and integrating from top to bottom, resulting in a volume of 0 when L = 0.
  • #1
Kqwert
160
3

Homework Statement


Peter has a spherical shaped water tank with radius R. At the top of the tank there's a small hole. Peter wants to know how much water there is left in the tank by measuring the distance L from the hole to the water surface.
Find an explicit form for the water volume V(L), 0 <= L <= 2R

Homework Equations

The Attempt at a Solution


So,

I considered the semi-circle
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which I then rotated around the x-axis,
i.e.
gif.gif


But this yields the wrong answer, in the solution manual they have the following solution:
gif.gif


Why is my solution wrong? I am trying to do the exact same thing as the solution proposes, just different integration limits.
 

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  • #2
What is the sum of your V and that from the solution ?
 
  • #3
Mine is:
gif.gif


Solution is:

gif.gif
 

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  • #4
@Kqwert, you didn't answer the question that @BvU asked.
BvU said:
What is the sum of your V and that from the solution ?
 
  • #5
Not sure if I understand the question? Do you mean the two expressions 'I've given in post #3 added together?
 
  • #6
Yes, of course :smile:. Add them up and be surprised.
 
  • Like
Likes SammyS
  • #7
uhm, it gives the volume of a sphere? How does that help me?
 
  • #8
So together you integrate from top to bottom.
What is the volume in the tank when L = 0 ? (Your integral would yield 0 in that case)
 

What is the Solid of Revolution HW and what does it mean to "solve" it?

The Solid of Revolution HW refers to a homework problem that involves finding the volume of a solid object that is created by rotating a 2-dimensional shape around a given axis. "Solving" this problem means finding the exact value of the volume, typically denoted as V(L), for a given range of values for the variable L.

What is the significance of the range 0<=L<=2R in this problem?

The range 0<=L<=2R represents the interval of values for the variable L that determine the volume of the solid object. In this case, L represents the distance from the axis of rotation to the edge of the 2-dimensional shape. The lower limit of 0 indicates that the shape starts at the axis, and the upper limit of 2R indicates that the shape extends to a maximum distance of 2 times the radius (R) of the shape.

How do you find the volume of the solid object for a specific value of L?

The volume of the solid object can be found using the formula V(L) = π * ∫(f(x))^2 dx, where f(x) represents the equation of the 2-dimensional shape and the integral is evaluated from 0 to L. This formula is based on the disk method or the washer method, depending on the shape of the cross-section of the solid object.

What types of 2-dimensional shapes can be used in this problem?

Any 2-dimensional shape can be used as long as it meets certain criteria. The shape must be continuous, have a defined equation, and be able to rotate around the given axis without overlapping itself. Common shapes used in this type of problem include circles, squares, triangles, and more complex curves.

Why is it important to "solve" the Solid of Revolution HW in real-world applications?

Finding the volume of a solid object created by rotating a 2-dimensional shape around an axis is a common application in various fields of science and engineering. It allows for the determination of the volume of objects such as cones, cylinders, and spheres, which are commonly found in nature and used in technology. It also provides a way to calculate the amount of material needed to manufacture these objects.

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