Please help with this Calculus 2 application integration problem(need help understanding the concept)(video)

Question

A tank is full of water. Find the work W required to pump the water out of the spout. Use the fact that water weighs 62.5 lb/ft³.

W = ft-lb

a= 2

b= 3

c = 4

Answer 1

Notice that the end face of the diagram is a rectangle 3 ft wide by 2 ft high. 

The origin of coordinates (x,y,z) can be chosen as the lower left corner of the rectangle.  

Let z be the vertical coordinate with z = 0 at the lowest point and z = 2 at the highest point. The z axis contains the left edge of the rectangle.

Let y = the coordinate that contains the other edge of the rectangle.   0 < y < 3

The x coordinate is perpendicular to the z and y coordinates.  0 < x  < 4.

Of the 5 surfaces of the volume, two are normal  to the y axis, one (the top) is normal to the z axis, and one ( the end face) is normal to the x axis. At any point on the fifth surface, it can be shown ( by similar triangles) that x = 2z 

The volume can be thought of as a stack of slabs of infinitesimal height dz and area   = 3 x = 6 z .

The associated infinitesimal volume is  6 z dz. The (infinitesimal)  weight of this infinitesimal volume is

 rho 6 z dz.   Here the density rho = 62.5.   To exit the spout , this weight must be raised up by 2 -z (feet).

This means that the total work is the integral of  (2-z) rho 6 z dz, with limits of 0 and 2.

The indefinite  integral is easy to carry out =  rho (6 z^2 – 2 z^3 )  

There is no contribution from the lower limit so the integral is rho (24 -16)

Which works out to be 500 lb-ft

Answer 2

If we take origin at vertex of tank - prizm and make section at any point with coordinate y perpendicular to y-axis, we get in section rectangle b and yc/a. So elementary work dW = b·yc/a·w(a - y)dy;

So, work W = ∫₀^adW = ∫₀^abcw/ay(a - y)dy = bcw/a∫₀^a(ay - y²)dy = bcw/a(ay²/2 - y³/3)₀^a = bcw/a(a³/2 - a³/3) =

= a²bcw/6 = 2²·3·4·62.5/6 lb·ft = 500 lb·ft

Answer 3

Hey Angie, here's the link to a video I made that goes through the problem! One thing I forgot to mention at the end is that it's always a good idea to "sanity check" your answer by checking units. In this problem, your answer should have units of lb*ft, and our answer holds up to this check!

Best,

Abby

https://youtu.be/TVM-Q2kmQK0