# Thread: Volume As An Integral

1. ## Volume As An Integral

I have a hard time determining when to use the washer method and using cylindrical shells when finding the volume of a solid. And when to start with Dy, Or Dx.

For Example: Find The Volume of the solid generated by revoling the region bounded by y = $\displaystyle x^(1/2)$ or the square root of x and the lines y = 2 and x = 0 about the:

A) the x- axis.

Would the washer method be more useful here?

2. Usually which depends which method is easier to take the integral of one could produce a rather difficult integral and one can provide a much simpler one, when doing these volume of revolution problems you partition the area by sections that are perpendicular to the axis of revolution you cut section that are perpendicular to the x-axis in your case so you will have a thickness of dx, because your partition are happening are happening when values of x change in this case. so what that means you need to get your function in terms of x which they usually are in a manner like this
$\displaystyle y=f(x)$

3. Originally Posted by RockHard
Usually which depends which method is easier to take the integral of one could produce a rather difficult integral and one can provide a much simpler one, when doing these volume of revolution problems you partition the area by sections that are perpendicular to the axis of revolution you cut section that are perpendicular to the x-axis in your case so you will have a thickness of dx, because your partition are happening are happening when values of x change in this case. so what that means you need to get your function in terms of x which they usually are in a manner like this
$\displaystyle y=f(x)$
i see that, but lets say you want to use either Dx or Dy, and the axis of revolution is shifted to, lets say, y = 2 or x = 2, what of the gap between the graphs?

4. As I stated you usually make partitions perpendicular to the x-axis, so it is not a matter of what you want to use, a graph revolution around any line y is the same as being rotated about the x - axis and vice versa If you have a "gap" between graphs you take in the outer radius^2 - inner radius^2 in the disk method, perhaps this link could explain better than I could.

Pauls Online Notes : Calculus I - Volumes of Solids of Revolution / Method of Rings
Pauls Online Notes : Calculus I - Volumes of Solids of Revolution/Method of Cylinder

What I did, as I do with an example just analyse the concept they applied in each example