Please verify my work - Calculus Problem: Work required to move liquid out of a tank?

Question

Hey all,

Thanks for helping me out with this. I already finished my Calculus 2 course, but I missed a problem similar to this on one of my tests and it just bugged me.

A cylindrical tank of radius 5 ft and height 9 ft is two-thirds filled with water. Find the work required to pump all the water over the upper rim.

From a previous problem, the given density of water is: 62.4 lb/cubic foot.

Each "k" in my equations is actually a subscript; just thought I'd mention that.

Wk = (Fk)(x*k)

Wk approximately equal to (Fk)(x*k); Fk = Force required to lift kth layer and x*k = distance of kth later from upper rim of cylinder.

So... I went with the force required to lift the kth layer = the weight of the kth layer.

Approximated the volume of the kth layer:

[Note: (delta xk) = height of the kth layer]

V = (pi)(r^2)(delta xk)

V = (pi)(5^2)(delta xk)

V = (pi)(25)(delta xk)

Fk = (62.4)(25)(pi)(delta xk)

Fk = (1560)(pi)(delta xk)

So, using Wk = (Fk)(x*k):

Wk = (1560)(pi)(delta xk)(x*k)

I put the above equation into a Riemann sum... so in my notes it looks like:

Summation from k=1 to n of (1560)(pi)(x*k)(delta xk)

I took the limit of the Riemann sum as (delta xk) approaches zero.

Which gave me a definite integral:

Integral from 3 to 9 of (1560)(pi)(x)(dx)

[So my lower bound is 3; upper bound is 9. Wasn't sure if I did this right.]

From there I came up with:

(1560)(pi) times the definite integral from 3 to 9 of x dx

(1560)(pi) times ((x^2)/2) evaluated from 3 to 9

(1560)(pi)((81/2) - (9/2))

I came out with 176431.84 ft lb ...?

Thanks in advance.

Sorry if that was confusing.

Answer 1

Integral from 3 to 9 of (1560)(pi)(x)(dx)

the x there is actually the distance of such a partition to the destination (namely, the top of the tank.) The partition is constant in the expression anyway.

since your integral is from 3 to 9, then the integral is of the correct form.

Had you had your integral from 0 to 6, you should have (9-x) instead of x.

Otherwise, you are good to go.. . .

if you did not understand my reply:

YOU ARE CORRECT. :-)