Final answer:
The student's question is about calculating the volume between the surfaces z=3r² and z=4-r² using integration. By setting the equations equal, we find the intersection radius and perform the integral to get the volume in terms of π.
Step-by-step explanation:
The student's question involves finding the volume of a region bounded by the surfaces z=3r² and z=4-r². To find this volume, we consider the volume of a solid of revolution generated by revolving the area bounded by these curves around the z-axis. The given surfaces represent a paraboloid and an inverted paraboloid, respectively. Therefore, we must integrate the difference of the functions (that give the height of the solid at each radius r) with respect to r, considering the limits of integration where the two surfaces intersect.
First, set 3r² equal to 4-r² to find the intersection radius, which gives r=1. The volume V can then be found by integrating π(4-r² - 3r²)ρ² from r=0 to r=1. Performing this integration gives the volume V as π cubic units.