Final answer:
To find the volume of the solid generated by revolving the region, we will use the shell method.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by x=6√y, x=−5y, and y=2 about the x-axis, we will use the shell method. For the shell method, we integrate the area of the shell (a thin cylindrical shell) over the interval of y-values that define the region. The volume is given by the formula V = 2π ∫ (x*y) dy, where the integral is taken over the y-values that span the region.
To apply the shell method, we first need to express x in terms of y. From the given equations, we have x = 6√y and x = −5y. Setting them equal to each other, we get 6√y = −5y. Squaring both sides and solving for y, we find y = (25/36).
Next, we integrate the expression x*y over the interval of y-values from 0 to (25/36). Evaluating this integral will give us the volume of the solid.