Final Answer
The volume of the region under the surface (z = xy²) and above the area bounded by (x = y²) and (x + 3y = 4) is 12/5 cubic units.
Step-by-step explanation
To find the volume of the region, we first need to determine the limits of integration. Solving (x = y²) and (x + 3y = 4) simultaneously gives us the points of intersection. Substituting (x = y²) into (x + 3y = 4), we get (y² + 3y = 4), which gives (y = 1) and (y = -4). These values define the limits of integration for (y).
Now, for (x), we integrate from the parabola (x = y²) to the line (x + 3y = 4). The integral setup for the volume is given by:
V = \int_{-4}^{1}\int_{y²}^{4-3y}xy²,dx \,dy]
Solving this double integral gives the final answer. Performing the integration yields the volume (12/5) cubic units.
In essence, the volume is obtained by summing up infinitesimally small volumes under the surface (z = xy²) within the specified region. The limits of integration define the boundaries of this region, and the double integral captures the infinitesimal slices that compose the entire volume. The resulting value, (12/5) cubic units, represents the total volume of the region under consideration.