A trough is 2 feet long and 1 foot high. The vertical cross-section of the trough parallel to an end is shaped like the graph of $y=x^{10}$ 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 $l b f$ or $l b * f t$ for $f t-I b$.)
\[
\text { Work }=
\]

Step 1 ：The work required to pump the water out of the trough is equal to the weight of the water times the distance it needs to be moved. The weight of the water is given by its volume times the weight per unit volume, and the distance it needs to be moved is equal to the height of the trough minus the height of the water at a given point.

Step 2 ：We can calculate the volume of the water by integrating the area of the cross-section along the length of the trough. The area of the cross-section is given by the integral of \(y=x^{10}\) from \(x=-1\) to \(x=1\), and the length of the trough is 2 feet.

Step 3 ：The distance the water needs to be moved at a given point is 1 foot minus the height of the water at that point, which is given by \(y=x^{10}\).

Step 4 ：So, the work required to pump the water out of the trough is given by the integral of the weight of the water times the distance it needs to be moved, integrated over the length of the trough.

Step 5 ：Let's calculate this in Python. x = x, y = x**10, area = 2/11, volume = 4/11, weight = 248/11, distance = 1 - x**10, work = 4960/121.

Step 6 ：The work required to pump the water out of the trough is \(\frac{4960}{121}\) foot-pounds.

Step 7 ：Final Answer: The amount of work required to empty the trough by pumping the water over the top is \(\boxed{\frac{4960}{121} \text{ foot-pounds}}\).