To find the amount of work required to empty the trough, calculate the weight of the water and the height to which it is raised. Divide the trough into small vertical rectangular strips and calculate the volume and weight of each strip. Integrate the weight of water over the range of x=-1 to x=1 to find the total work.
To find the amount of work required to empty the trough, we need to calculate the weight of the water and the height to which it is raised. The vertical cross-section of the trough is shaped like the graph of y=x^2 from x=-1 to x=1. So, the trough can be divided into small vertical rectangular strips of width dx.
Let's consider one of these strips. The width (w) of the strip is dx, the height (h) of the strip is given by y=x^2, and the length (l) of the strip is 2 feet.
The volume of water in one strip is V = l * w * h = 2 * dx * x^2.
The weight of water is 62 pounds per cubic foot. So, the weight of water in one strip is 62 * V = 62 * 2 * dx * x^2.
The total work required to empty the trough is obtained by integrating the weight of water over the range of x=-1 to x=1. So, the total work is given by W = ∫[from -1 to 1] (62 * 2 * x^2) dx.
By evaluating the integral, we can find the exact amount of work required to empty the trough.