Consider the solid obtained by rotating the region bounded by the given curves about the y-axis.
y = ln x, y = 4, y = 5, x = 0
Find the volume V of this solid.
Help!!! Thank you in advance :(
This problem can be easily solved using the disk method.
Horizontal disks are used, with slices of thickness dy.
We will integrate from y=4 to y=5.
Each disk has a volume of πr(y)²dy.
where the radius is a function of y.
Since y=ln(x), its inverse relation is x=e^y.
Integrate for y=4 to 5 of
V=∫π(e^y)²dy
=π∫e^(2y)dy
=π(1/2)e^(2y)
Evaluate between 4 and 5 gives
V=(π/2)(e^(2*5)-e^(2*4))
=29917 (approx.)
Check:
The average radius is between e^4 and e^5=101.5
Volume = 32400 approx. > 29917
Since the curve ln(x) is concave up, the actual volume should be a little less than the approximation. So the calculated volume should be correct.
To find the volume of the solid obtained by rotating the region bounded by the given curves about the y-axis, we can use the method of cylindrical shells.
The cylindrical shell method involves integrating the volume of a series of cylinders that make up the solid.
1. First, let's sketch the region bounded by the curves y = ln x, y = 4, y = 5, and x = 0 on a coordinate plane.
The region is a horizontal strip that extends from y = 4 to y = 5 and is bounded by the curve y = ln x and the y-axis.
2. Now, let's consider a thin vertical strip of width Δy on the coordinate plane. This strip is located at a specific y-value between 4 and 5.
3. We can represent the height of this strip as y and the corresponding x-values as x_1 and x_2.
4. The volume of the cylindrical shell can be approximated as the product of the circumference of the shell (2πx) and the height of the shell (Δy).
5. The limits of integration for Δy will be from 4 to 5, as that represents the height of the region.
6. To find x_1 and x_2, we need to solve for x in terms of y in the equation y = ln x.
7. Rearranging the equation, we get x = e^y.
8. Now, let's write the integral to calculate the volume V:
V = ∫[4,5] 2πx * Δy
9. Substitute x = e^y into the equation:
V = ∫[4,5] 2π(e^y) * Δy
10. Integrate the expression with respect to y:
V = 2π ∫[4,5] e^y * Δy
11. Evaluate the integral:
V = 2π [e^y] [4,5]
V = 2π * (e^5 - e^4)
12. Simplify the expression to find the final volume V.
V ≈ 384.62 cubic units (rounded to two decimal places)
Therefore, the volume of the solid obtained by rotating the region about the y-axis is approximately 384.62 cubic units.