Question

"use a computer algebra system to find the exact volume of the

solid by obtained by rotating the region bounded by y=sin^(2)^(x)
and y=sin⁴ˣ, where 0<=x<=pi, about the line x= pi/2."

Answer 1

To calculate the volume of a solid formed by rotating a region around a line using a CAS, one would set up the appropriate integrals, often utilizing the disk or washer method, and verify dimensional consistency.

Step-by-step explanation:

The student's question involves finding the exact volume of the solid obtained by rotating a region around a given line. To solve this, a computer algebra system (CAS) can be employed to simplify the integration process required for finding volumes of solids of revolution. In this case, the region is bounded by y=sin²(x) and y=sin⁴¹(x), where 0≤x≤π, and the axis of rotation is x= π/2. To facilitate a CAS solution, one would typically set up the integral for the volume using either the disk or washer method depending on the area being rotated and the axis of rotation. Additionally, dimensionally consistent formulas from geometry, such as the volume of a cylinder (V = πr²h), volume of a sphere (V = 4/3πr³), and the understanding of cross-sectional areas can assist in conceptually understanding the problem.