Work Required to Empty a Hemispherical Tank

[Precursor]

Homework Statement
Find the work required to empty a tank in the shape of a hemisphere of radius $$R$$ meters through an outlet at the top of the tank. The density of water is $$p kg/m^{3}$$; the acceleration of a free falling body is $$g$$. (Ignore the length of the outlet at the top.)


The attempt at a solution

$$w = \int_a^b (density)(gravity)(Area-of-slice)(distance)dx$$

$$w = \int_0^R (p)(g)(\pi)(R^{2})(R - x)dx$$

Is this correct/complete?

 

[tiny-tim]

Hi Precursor!

(have a pi: π and a rho: ρ and an integral: ∫ and try using the X² tag just above the Reply box )

Find the work required to empty a tank in the shape of a hemisphere of radius $$R$$ meters through an outlet at the top of the tank. The density of water is $$p kg/m^{3}$$; the acceleration of a free falling body is $$g$$. (Ignore the length of the outlet at the top.)
…
$$w = \int_0^R (p)(g)(\pi)(R^{2})(R - x)dx$$

Is this correct/complete?

No, that's the correct formulal for a cylinder (Area-of-slice = πr²).

Try again! ​

 

[Precursor]

So is the area of the slice actually π(1 - x²)?

 

[tiny-tim]

So is the area of the slice actually π(1 - x²)?

Nooo (btw, it might be easier if you measured x from the top instead of from the bottom )

 

[Precursor]

Is it π√(R² - x²)?

 

[tiny-tim]

Is it π√(R² - x²)?

With x is measured from the top, yes except …

lose the square-root!
​

(and I'm going to bed :zzz: g'night!)​