Question

Let R denote the finite region enclosed by the curves y = 1 and y = x²

. Let S denote the solid obtained by rotating R about y-axis. We are interested the volume of the solid S : call this volume V.
(a) Make a good sketch of the region R.

Answer 1

The student's question pertains to the volume of a solid obtained by revolving a finite region around the y-axis, which can be calculated using the method of disks or washers in calculus. The volume of a sphere, as related to the question, is given by ⅔πr³, distinguishing it from its surface area, 4πr².

Step-by-step explanation:

The subject of the student's question is Mathematics, specifically involving calculus and the concept of volumes of solids of revolution. When the problem talks about rotating the region R, enclosed by the curves y = 1 and y = x², around the y-axis to obtain a solid S, it's referring to a method called the disk or washer method to find the volume of the resulting solid.

Understanding the Formula for Volume

Referencing the materials provided, the formula for the volume of a sphere is given by V = ⅓πr³, which comes from the equation for a sphere's volume V = ⅔πr³. The expression 4πr² represents the surface area of the sphere. Therefore, the volume of a sphere of radius r is correctly represented by the ⅔πr³ term. This conceptual understanding aids in remembering and deriving the formulas for various solids.