Parabolic Water Tank Homework: Find Work W Done to Lower Water

[amw2829]

Homework Statement

The ends of a "parabolic" water tank are the shape of the region inside the graph of
y = x2
for
0 ≤ y ≤ 4
; the cross sections parallel to the top of the tank (and the ground) are rectangles. At its center the tank is 4 feet deep and 4 feet across. The tank is 6 feet long. Rain has filled the tank and water is removed by pumping it up to a spout that is 4 feet above the top of the tank. Set up a definite integral to find the work W that is done to lower the water to a depth of 3 feet and then find the work. [Hint: You will need to integrate with respect to y.

Homework Equations

W=62.5(l-x)(A(x)) dx

The Attempt at a Solution

Since you need to integrate with respect to y, I would assume the integral would be: 62.5(6-y)(A(y)). However, I'm not sure what A(y) would be or how I would identify the bounds.

 

[HallsofIvy]

A(y) is the area of the rectangular cross section of the tank at height y. One side of that rectangle has length 6. The other side has length 2x where y= x^2.