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.

... 36. Set up the integral for the surface area of the surface of revolution, and approximate the integral with a ... 18. Compute the volume of the solid formed by ...

... Several volumes of solids of revolution are calculated, and ... we to ﬁnd the volume of the solid obtained by ... The volume is given by the integral √ √ √ 8 8 ...

Mathematics. Find volume of the solid of revolution formed when the region bounded by y=sinx and the x axis for 0 < or=z< or =pi is revolved about the x ...

... the graphs of y = x and Figure 20.1 is rotated around the line y = x. Find (a) the centroid of the region and (b) the volume of the solid of revolution. ...

... 2. Water/sewage 3. Energy 4. Solid waste 5 ... of materials won't come judged at volumes similar to ... of a To order Costing the Ultralite in Volume Production: Can ...

... most of which touts the "unlimited" possibilities of this "revolution" in education. ... "Start with a solid foundation and then ... The details can speak volumes.". ? ...

... 18 In earlier editions of this volume, using data ... of most boards, usually vote as a solid, unified block ... Following the Industrial Revolution in America in the ...

... The volumes of solids of revolution are found using the shell... 2. The surface area of the solid is A = 2π ... height ) = 2π ( x)(1 − x) , so the volume is 1 ...

...Volumes of solids of revolution are found using the disk and ...method of finding the volume of a solid, by rotation ... In the shell method, the volume of a typical ...