Finding the volume of unbounded solids of revolution

(a) Find the volume of the unbounded solid generated by rotating the unbounded region of y=e^(-x) with x>=1 around the x axis (see the attached figure)
(b) What happens if y=1/sqrt(x) instead?

The volume of the solid of revolution obtained by ...

Solution Summary

Rotating a bounded portion of the x axis of a graph about the x axis yields "Solids of Revolution". Integration can be used to find the volume of the solids so formed. When the x region is not bounded, we can still perform the rotation to give an *unbounded* solid of revolution. We can attempt to find a volume for this object by first finding the volume of the bounded object in terms of the right hand bound, then trying to take the limit as the right hand bound goes to infinity. This limit may or may not exist. The solution comprises a 1/2 page word attachment with equations written in Mathtype showing two examples of this technique, 1 for which the volume exists and one for which it does not. Step by step explanation is given.

Volume of solids of revolution..
1. A paraboloid dish (cross section ) is 8 units deep. It is filled with water up to a height of 4 units. How much water must be added to the dish to fill it completely?
4. Write an integral that represents thevolume of the solid formed by rotating the region bounded by , , , and

Find thevolume of the solid that is generated by rotating around the indicated axis the plane region bounded by the fiven curves.
1) y=√x,y=0,x=4; The x-axis
2) y= 1/x, y=0, x=0.1,x=1; the x-axis
3)Find thevolume of the ellipsoid generated by rotating around the x-axis the region bounded by the ellipse with equation.

Find thevolume of the solid formed by revolving the region bounded by y=x^3, x=2, and y=1 about the y-axis.
keywords: integrals, integration, integrate, integrated, integrating, double, triple, multiple

Question (1)
What is thevolume of the solid of revolution obtained by rotating the region bounded by y = 1 and y = 5 - x^2 around the X-axis.
Question (2)
Find thevolume of the solid of revolution obtained by rotating the region bounded by y = 1 and y = Tan x about the x-axis from x = 0 to x = pi/4.
See attached file

Find the volume of the unbounded solid generated by rotating the unbounded region around the x axis. This is the region between the graph of y= e^-x and the x axis for x> or= 1. [Method: Compute the volume from x=1 to x=b, where b> 1. Then find the limit of this volume as b -->+infinity] What about if y=1/SQRTx.

A)The region bounded by y = e^2x, y = e^x, x = 0, and x = 2 is rotated around the x-axis. Find thevolume.
b)Consider the region bounded by y = x^2, y = 3, and the y-axis. Find thevolume of the solid obtained by rotating the region around the y-axis.

Please help with the following problems. Please provide step by step calculations and diagrams.
For the following problem, sketch the region and then find thevolume of the solid where the base is the given region and which has the property that each cross-section perpendicular to the x-axis is a semicircle.
1) the regio

Problem: The region R is bounded by the graphs of x - 2y = 3 and x = y2. Find the integral that gives thevolume of the solid obtained by rotating R around the line x = -1.
I'm having a hard time setting up the integral, I think that I have the concept for findingthe area of a 2d object using an integral but can't figure out