Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves y=1/x, y=0, x=1 and x=4
about the line y=−1
v = ∫[1,4] π(R^2-r^2) dx
where R=(1/x + 1) and r=1
v = ∫[1,4] π((1/x + 1)^2-1^2) dx = π(3/4 + ln16)
check, using shells of thickness dy:
v = ∫[0,1/4] 2πrh dy
where r=y+1 and h=3
+ ∫[1/4,1] 2πrh dy
where r=y+1 and h=(1/y - 1)
Crank it out, and you get the same answer.
Well, well, well! Let's see what we have here. We have a region bounded by curves, rotating around a line. Sounds like a math party!
To find the volume of this solid, we can use the method of disks or washers. Since we're rotating around the line y = -1, let's consider each slice of the solid as a disk.
The radius of each disk will be the distance between the line y = -1 and the curve y = 1/x. We can express this distance as:
r = (1/x) - (-1) = (1+1/x)
Now, to find the area of each disk, we need to square the radius, multiply by π and integrate it over our bounded region.
But wait! The curve y = 1/x has a vertical asymptote at x = 0, which means we can't rotate the area from 0 to 1. So we'll have to integrate from 1 to 4 and then double it (since the same area is on both sides of y = -1).
So, get ready for some integrals, my friend!
The volume V can be found by integrating (π * r^2) dx from x = 1 to x = 4, and then multiplying the result by 2:
V = 2 * ∫[1,4] (π * (1+1/x)^2) dx
Now, if you're not afraid of some math circus, go ahead and evaluate that integral! But beware, it might get a little messy.
To find the volume of the solid obtained by rotating the region bounded by the curves y=1/x, y=0, x=1, and x=4 about the line y=−1, we can use the method of cylindrical shells.
Step 1:
First, let's draw a rough sketch of the region to visualize it.
The region is bounded by the curves y=1/x, y=0, x=1, and x=4.
Step 2:
Next, we need to find the height of the cylindrical shells. The height will be the difference in y-values between the curve y=1/x and the line y=−1.
Since we're rotating about the line y=−1, the height will be given by:
height = (1/x) - (-1)
Simplifying this, we get:
height = (1/x) + 1
Step 3:
Now, let's find the radius of the cylindrical shells. The radius will be the difference in x-values between the boundaries x=1 and x=4.
radius = 4 - 1 = 3
Step 4:
The volume of each cylindrical shell can be calculated using the formula:
volume = 2π(radius)(height)(thickness)
In this case, the thickness of each shell is dx, since we are integrating with respect to x.
Therefore, the volume of each shell is:
dV = 2π(3)(1/x + 1)(dx)
Step 5:
To find the total volume, we need to integrate the expression for dV over the interval [1, 4].
∫(1 to 4) 2π(3)(1/x + 1) dx
This integral can be split into two separate integrals:
∫(1 to 4) 2π(3)(1/x) dx + ∫(1 to 4) 2π(3)(1) dx
Step 6:
Integrating the first integral:
∫(1 to 4) 2π(3)(1/x) dx = 2π(3) * ln|x| |(1 to 4)
= 2π(3)(ln(4) - ln(1))
= 6π ln(4)
Step 7:
Integrating the second integral:
∫(1 to 4) 2π(3)(1) dx = 2π(3) * x |(1 to 4)
= 2π(3)(4 - 1)
= 6π(3)
= 18π
Step 8:
Adding the results of the two integrals together:
6π ln(4) + 18π
= 6π (ln(4) + 3)
Hence, the volume of the solid obtained by rotating the region bounded by the curves y=1/x, y=0, x=1, and x=4 about the line y=−1 is 6π (ln(4) + 3).
To find the volume of the solid obtained by rotating the given region about the line y = -1, we can use the method of disks or washers.
First, let's sketch the region bounded by the curves.
The curves y = 1/x and y = 0 intersect at x = 1 and x = 4. This means that our region is bounded by x = 1 and x = 4.
Now, let's consider a small strip of width ∆x within this region, located at an arbitrary x-value between 1 and 4.
The height of this strip can be found by subtracting the y-values of the curves at that particular x-value. Since the line y = -1 is the axis of rotation, the distance from the strip to the axis of rotation is 1 unit.
Therefore, the radius of the disk or washer at this x-value is 1 unit. The thickness of the disk or washer, ∆x, remains constant.
The volume of each disk or washer can be approximated by treating it as a cylindrical solid. The volume of a cylindrical solid can be calculated as V = π * r^2 * h, where r is the radius and h is the height.
In this case, the radius of the disk or washer is always 1 unit, and the height of each disk or washer is the difference between the y-values of the curves.
So, for each small strip of width ∆x, the volume of the corresponding disk or washer is approximately V = π * (1)^2 * (∆y).
To find the total volume, we need to sum up the volumes of all these disks or washers. This can be done by integrating the expression for the volume over the interval [1, 4].
Therefore, the volume of the solid obtained by rotating the region bounded by the curves y = 1/x, y = 0, x = 1, and x = 4 about the line y = -1 is given by the integral:
V = ∫[1, 4] (π * (1)^2 * (∆y))
To evaluate this integral, we need to express ∆y in terms of x. From the given curves, we observe that ∆y = 1/x - 0. Therefore, we can rewrite the integral as:
V = ∫[1, 4] (π * (1)^2 * (1/x)) dx
Now, we can evaluate this integral to find the exact volume of the solid.