use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. sketch the region and a typical shell.
x=1+y^2, x=0, y=1, y=2
The region is roughly trapezoid shaped, with vertices at (0,1)(0,2)(2,1)(5,2)
With shells, we have to integrate on x, since the shell thickness is dx.
From x=0-2, we just have a rectangle (which, revolved is just a cylinder of radius 2, height 1), and for x=2-5, we have shells of height 2-y.
Since x=1+y^2, y = √(x-1)
v = 2π*2*1 + ∫[2,5] 2πrh dx
where r = x and h = 2-√(x-1)
v = 4π + 2π∫[2,5] x(2-√(x-1)) dx
= 4π + 2π(59/15)
= 178/15 π
just to check, we can use discs, and we have
v = ∫[1,2] πr^2 dy
where r = x
v = π∫[1,2] (y^2+1)^2 dy
= 178/15 π
just one more question how did you get the y=squareroot of x-1?? so confuse
x = 1+y^2
x-1 = y^2
√(x-1) = y
Algebra I, man, Algebra I.
To find the volume of the solid obtained by rotating the region bounded by the curves x = 1 + y^2, x = 0, y = 1, and y = 2 about the x-axis using the method of cylindrical shells, follow these steps:
Step 1: Plotting the region
To better understand the problem, let's start by plotting the region bounded by the given curves. On a Cartesian coordinate system, plot the curve x = 1 + y^2, along with the lines x = 0, y = 1, and y = 2. This will help provide a visual representation of the region we need to rotate.
Step 2: Identifying the region
Looking at the graph, you can see that the region lies between the curves x = 1 + y^2 and x = 0, and is bounded by the lines y = 1 and y = 2.
Step 3: Setting up the integral for volume
To calculate the volume using cylindrical shells, imagine taking thin, vertical strips (shells) along the x-axis, perpendicular to it. The volume of each shell can be calculated by multiplying its height (the length of the strip) by its circumference.
Step 4: Determining the height and circumference of each shell
The height of each shell can be found by taking the difference between the two curves at each x-coordinate. In this case, the height can be expressed as (2 - 1), or simply 1.
The circumference of each shell can be calculated by finding the distance around the curve at each x-coordinate. The curve x = 1 + y^2 can be rewritten as y = sqrt(x - 1). Therefore, the circumference for each shell can be expressed as 2π * radius, where the radius is y.
Step 5: Determining the limits of integration
To find the limits of integration, we need to determine the x-values at which the curves intersect. In this case, we have x = 1 + y^2 and x = 0. By setting the two equations equal to each other, we can solve for the point of intersection:
1 + y^2 = 0
y^2 = -1 (Not possible for real numbers)
Since there is no intersection point, the limits of integration for the volume integral will be from y = 1 to y = 2.
Step 6: Calculating the volume
Now that we have all the necessary information, we can set up the volume integral:
V = ∫[1, 2] (2π * y) * 1 dy
Integrating with respect to y, we get:
V = 2π ∫[1, 2] y dy
Evaluating the integral, we have:
V = 2π * [0.5y^2] [1, 2]
V = 2π * (0.5(2^2) - 0.5(1^2))
V = 2π * (2 - 0.5)
V = 2π * 1.5
V = 3π cubic units
Therefore, the volume of the solid obtained by rotating the region bounded by the curves x = 1 + y^2, x = 0, y = 1, and y = 2 about the x-axis using the method of cylindrical shells is 3π cubic units.