Final answer:
The volume of the solid under the paraboloid z=x²+y² and above region D is found by performing a double integral within the bounds determined by y=2x and y=x², using the intersection points x=0 and x=2 as integral limits.
Step-by-step explanation:
To find the volume of the solid under the paraboloid z=x²+y² and above the region D in the xy-plane, we set up the integral bounds using the curves y=2x and y=x², which intersect where x²=2x, leading to x=0 and x=2 as the limits for x. For the bounds of y, we use the given relationships and express y in terms of x for each region.
Next, we perform a double integral where the outer integral (with respect to x) runs from 0 to 2, and the inner integral (with respect to y) runs from x² to 2x. The function to integrate is z=x²+y², representing the height of the paraboloid above the xy-plane at each point.
The volume V is then defined by the double integral:
V = ∫0∫2 dx ∫x²∫2x (x²+y²) dy
Calculating this double integral gives us the volume of the solid in question.