Question

Find the volumes of the solids generated by revolving the described region about the given axis:

The region enclosed above by the curve y=1+(x²)4, below by the x-axis, to the left by the y-axis, and to the right by the line x=3, rotated about the y-axis.

Answer 1

The volume of the solid can be determined by the cylindrical shells method, integrating the circumference of shells multiplied by their height and thickness from x=0 to x=3 for the region bound by y=1+(x²)³, the x-axis, and the line x=3, rotated about the y-axis.

Step-by-step explanation:

The student is asking about finding the volume of a solid generated by revolving a region around the y-axis. The region is bound by the curve y = 1 + (x²)³, the x-axis, the y-axis, and the line x = 3. This problem involves calculating the volume using the method of cylindrical shells or the disk method, where the volume is determined by integrating the area of infinitesimally thin disks or cylindrical shells that make up the solid.

To calculate the volume of the solid, we can use the cylindrical shells method, which involves integrating the circumference of the shell (2πx) times the height of the shell (y - 0, since the lower boundary is the x-axis) times the thickness of the shell (dx).

The volume V is represented by the integral from 0 to 3 of 2πx(1 + (x²)³)dx. Evaluating this integral will give us the exact volume of the solid when it is rotated about the y-axis.