A trough is 3 feet long and $1$ foot high. The vertical cross-section of the trough parallel to an end is shaped like the graph of $y=x^{6}$ from $x=-1$ to $x=1$. The trough is full of water. Find the amount of work required to empty the trough by pumping the water over the top. Note: The weight of water is 62 pounds per cubic foot. Your answer must include the correct units. (You may enter lbf or lb*ft for ft-lb.)

Work =