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

[Kqwert]

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

which I then rotated around the x-axis,
i.e.

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

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

 

[BvU]

What is the sum of your V and that from the solution ?

 

[Kqwert]

Mine is:

Solution is:

 

[Mark44]

@Kqwert, you didn't answer the question that @BvU asked.

What is the sum of your V and that from the solution ?

 

[Kqwert]

Not sure if I understand the question? Do you mean the two expressions 'I've given in post #3 added together?

 

[BvU]

Yes, of course . Add them up and be surprised.

 

[Kqwert]

uhm, it gives the volume of a sphere? How does that help me?

 

[BvU]

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)