Hello! I have a problem on volumes but I am having trouble visualizing the object.
y=ln(x), y=1, y=2, x=0, about the y-axis
For the graph, I graphed y=ln(x), then I have from 1 to 2. Is this one with a washer where I have integral fom 1 to 2 of pi*ln(x)^2 - pi* something?
I'm not so sure where to go after graphing it. It looks like an undefined area because there is no x-limit.
Thanks for your help.
The limit of x is determined by y=2 (y= ln(x) ) or x= e^2 at end
I would have integrated the volume as dy*PI radius^2 where radius is rad=e^y
check my thinking.
Hello! It seems like you are trying to find the volume of the solid generated by revolving the region bounded by the curve y=ln(x), the lines y=1 and y=2, and the y-axis about the y-axis. I'll guide you through the steps to get the answer.
First, let's consider the region bounded by the given curves. From the graph, you correctly identified the limits of y, which are y=1 and y=2. Now, we need to find the corresponding x-values for these limits.
To do that, we can solve the equation y=ln(x) for x. Since ln is the natural logarithm with base e, we have:
e^y = x
Now, substitute the y-values into this equation to find the x-values:
When y=1, x=e^1=e
When y=2, x=e^2
So, the x-values corresponding to the limits of y are x=e and x=e^2.
Now, let's think about the method to calculate the volume. Since we are rotating the region about the y-axis, we need to use the method of cylindrical shells.
Each cylindrical shell has a height of dy, a radius of r, and a thickness of dx (or dy, in this case, since we are integrating with respect to y). The radius of each shell is the distance of the point on the curve y=ln(x) from the y-axis, which is given by r=e^y.
The volume of each shell is given by the formula: dV = 2πrhdy, where r is the radius and h is the height.
Now, we can set up the integral to find the total volume:
V = ∫[from y=1 to y=2] (2πe^y)dy
Simplifying and integrating, we get:
V = 2π ∫[from y=1 to y=2] (e^y)dy
Integrating e^y gives us e^y, so the integral becomes:
V = 2π [e^y] [from y=1 to y=2]
V = 2π (e^2 - e)
Therefore, the volume of the solid generated by revolving the given region about the y-axis is 2π (e^2 - e).
I hope this explanation helps you understand the process of finding the volume in this problem! Let me know if you have any further questions.