Final answer:
To set up the iterated integral for the volume under the graph Z = 4x² + cos(y), solve the system of equations for the region's boundaries, then integrate Z over the region with x as a function of y.
Step-by-step explanation:
To propose an iterated integral for computing the volume under the graph of Z = 4x² + cos(y) on the given region, we must first find the intersection of the curves defined by the equations x = y² - 2 and x - 2y = 1.
Once the intersection points are calculated (which involves solving the system of equations for x and y), these points will provide the limits for the integral with respect to y. After that, we use x = y² - 2 and x = 2y + 1 to define the limits for x, in terms of y.
The volume under the surface Z can be found by integrating the function Z over the region defined by the boundaries of x in terms of y. The iterated integral will generally take the form:
∫∫ (4x² + cos(y)) dx dy,
where the outer integral is with respect to y, from the lower to the upper limit found from the intersection points, and the inner integral is with respect to x, between the curves x = y² - 2 (as a lower limit) and x = 2y + 1 (as an upper limit).