A trough is 7 feet long and 1 foot high The vertical crossse

A trough is 7 feet long and 1 foot high. The vertical cross-sections of the trough parallel to its ends are shaped like the graph offromto. The trough is completely filled with water. Find the amount of work in foot-pounds required to empty the trough by pumping the water over the top. Note: The weight of water ispounds per cubic foot. Work =ft-lb

A trough is 7 feet long and 1 foot high. The vertical cross-sections of the trough parallel to its ends are shaped like the graph of y=x^{8} from x=-1 tox=1. The trough is completely filled with water. Find the amount of work in foot-pounds required to empty the trough by pumping the water over the top. Note: The weight of water is 62.5 pounds per cubic foot. Work =ft-lb

Solution

as by converting the power we have x=y^(1/8). So, the volume of the solid rectangle of water is L*W*H. L=7 W=2*y^(1/8) H=dy (the infinitesimal height of the rectangular slice) so, the work for a single slice is 7*2*62*(y^[1/8])*(1-y)dy the resulting unit is ft-lbf (ft*ft*ft*lbf/ft^3*ft) now we integrate for y=0 to y=1 868 * integral (y^[1/8]-y^[5/8]) evaluated 0 to 1 868 * {8/5 y^(5/8) - 4/9 y^(9/8) }evaluated 0 to 1 868 * {1.6 - 0.62 - 0 -0} 868ft-lbf