find the answer in foot-pounds

Question

A trough is 5 feet long and 1 foot high. The vertical cross-section of the trough parallel to an end is shaped like the graph of  from  to  . The trough is full of water. Find the amount of work in foot-pounds required to empty the trough by pumping the water over the top. Note: In this problem, use 62 pounds per cubic foot as the weight of water.

Answer 1

Since y=x¹⁰ then x=y^1/10 so the width of the side is 2x =2y^1/10 and the thickness of a rectangular slice is dy. So the area of the vertical section is 2y^1/10dy and the volume of the trough is 10y^1/10dy. So the mass is 62.410y^1/10dy or mass = 624y^1/10dy. Want to empty the trough so need to move water a height 1-y ft. So work becomes 624∫₀¹(1-y)y^1/10dy. Integrating this gives 624(y^11/10/11/10 - y^21/10/21/10)₀¹ which gives 624( 10/11-10/21) and this gives 1890.91 ft-lbs.