Give the volume of the solid generated by revolving the region bounded by the graph of y=ln(x), the x-axis, the lines x=1 and x=e, about the y-axis
I am going to do this in a bit of a tricky way.
I can foresee having to integrate (ln(x))^2 which could be rather messy
so let's take the inverse of the whole mess
If we had:
Give the volume of the solid generated by revolving the region bounded by the graph of y=e^x, the y-axis, the lines y=1 and y=e, about the x-axis,
we would have exactly the same solid.
Volume = pi[integral](e^2 - (e^x)^2)dx from x=0 to x=1
= pi[(e^2)x - (1/2)e^(2x)] from 0 to 1
= pi[e^2 - (1/2)e^2 - (0 - (1/2)e^0))]
= (1/2)_e^2 - 1/2 or (e^2 - 1)/2
check my arithmetic.
To find the volume of the solid generated by revolving the region bounded by the graph of y = ln(x), the x-axis, the lines x = 1, and x = e about the y-axis, we can use the method of cylindrical shells.
Here's a step-by-step explanation of the process:
1. First, let's sketch the graph of y = ln(x) to visualize the region we need to revolve. The graph of ln(x) is an increasing curve that passes through the point (1, 0) and approaches positive infinity as x goes to infinity. The other boundary points are x = 1 and x = e, where e ≈ 2.71828.
2. Since we are revolving the region about the y-axis, we need to express the curve in terms of y rather than x. By rearranging the equation y = ln(x), we have x = e^y.
3. Now, we need to determine the limits of integration for the volume calculation. In this case, y ranges from 0 to ln(e).
4. Next, we consider an infinitesimally thin strip at a specific value of y. The thickness of this strip is denoted as Δy.
5. To find the height of the shell at a given y-value, we subtract the lower boundary (x = 1) from the upper boundary (x = e^y). So the height of the shell is e^y - 1.
6. The circumference of the shell is calculated by multiplying the height by 2π.
7. Finally, the volume of each shell is given by the product of its height, circumference, and thickness. We integrate this expression from y = 0 to y = ln(e) to sum up all the individual volumes.
To summarize, the volume V can be obtained using the following integral:
V = ∫[0, ln(e)] (2π * (e^y - 1) * Δy)
By evaluating this integral, you can find the volume of the solid generated by revolving the region bounded by the graph of y = ln(x), the x-axis, the lines x = 1, and x = e about the y-axis.