Approximate the volume of the solid generated by revolving region formed by the curve y=x^2, x-axis and the line x=2.
Volume approximated by concentric shells
a) Sketch the reqion y=x^2, x-axis and the line x=2.
b) We'll approximate the volume revolving the region about the y-axis.
c) partition the interval [0, 2) in x, so the width of each sub-interval..
d) On your sketch in part a, sketch a typical rectangle. What is the base of the rectangle? What is the height of the rectangle? (Since the base of the rectangle is & then the height must be a function of x.)
e) Sketch the solid you get by revolving this rectangle about the y-axis. What is the radius of this shell? What is the height of the shell? What is the width (thickness) of the shell?
f) Write a formula for the volume of the typical shell.
g) Write the Riemann sum that corresponds to adding up the volumes of n such shells.
h) Write the definite integral that is the limit of the Riemann sums as the number of shells increases without bound.
i) Evaluate the integral to calculate the exact volume of the solid.
First the region is sketched, shell height is defined, Riemann sums ...
The Volume of a Solid of Revolution is found by the Shell Method