Final answer:
To find the volumes, we use the disk method for rotation about the x-axis, and the shell method for rotation about the y-axis. The calculations involve setting up and evaluating definite integrals that depend on the rotation axis.
Step-by-step explanation:
Finding the Volume of a Solid of Revolution
To find the volume of the resulting solid when the region under the curve y = 6/(x^2 + 5x + 6) from x = 0 to x = 1 is rotated about the x-axis, we use the disk method. The volume V can be found by integrating the cross-sectional area A(x) of the disk perpendicular to the x-axis from x=0 to x=1. The cross-sectional area A(x) is π[y(x)]^2.
Solution for (a): Rotating about the x-axis gives us V = ∫01 A(x)dx = π ∫01 [6/(x^2 + 5x + 6)]^2 dx.
For rotation about the y-axis, the shell method needs to be used since we are revolving around the y-axis. To set this up, we consider the volume of each cylindrical shell with radius x, height y(x), and thickness dx. The volume V is therefore given by V = ∫01 2πxy dx.
Solution for (b): Rotating about the y-axis gives us V = 2π ∫01 x[6/(x^2 + 5x + 6)] dx.