The region R is bounded by the x-axis, x = 2, and y = x^2. What is the the volume of the solid formed by revolving R about the line x = 2?
y=x^2 and x=2 intersect at (2,4)
the radius of the rotated region = 2 - x
= 2 - √y
V = π∫ (2-√y)^2 dy from 0 to 4
= π∫ (4 - 4√y + y) dy from 0 to 4
= π[ 4y - (8/3)y^(3/2) + (1/2)y^2 ] from 0 to 4
= π (16 - (8/3)(8) + 8 - 0)
= 8π/3
check my arithmetic and thinking.
y=x^2 intersects x=2 at (2,4)
using shells,
v = ∫2πrh dx [0,2]
where
r = 2-x
h = y
v = 2π∫(2-x)(x^2)dx [0,2]
= 2π∫2x^2 - x^3 dx [0,2]
= 2π(2/3 x^3 - 1/4 x^4)[0,2]
= 2π(16/3 - 4)
= 2π(4/3)
= 8π/3
using discs,
v = ∫πr^2 dy [0,4]
= π∫(2-√y)^2 dy [0,4]
= π∫(4 - 4√y + y)dy [0,4]
= π(4x - 8/3y√y + 1/2 y^2)[0,4]
= π(16 - 64/3 + 8)
= 8π/3
To find the volume of the solid formed by revolving region R about the line x = 2, we will use the method of cylindrical shells.
Step 1: Graph the region R
To understand the region R, let's determine where the curve y = x^2 intersects the x-axis and the line x = 2.
Setting y = 0, we can solve the equation x^2 = 0. The only solution is x = 0.
So, region R is a semi-circle bounded by the x-axis, x = 2, and y = x^2. It looks like this:
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Step 2: Determine the height of the cylindrical shell
The height of the cylindrical shell will be the difference between the y-coordinate of the curve y = x^2 and the y-coordinate of the x-axis.
The equation of the curve y = x^2 can be rewritten as x = sqrt(y).
At x = 2, the y-coordinate can be found by substituting x = 2 into the equation y = x^2:
y = (2)^2 = 4.
So, the height of each cylindrical shell is the difference between y = x^2 and the x-axis, which is y = 0. Therefore, the height is 4 - 0 = 4.
Step 3: Determine the radius of the cylindrical shell
The radius of each cylindrical shell is the distance between the vertical line x = 2 and the x-coordinate of the curve y = x^2.
Since the line x = 2 is a vertical line, its distance from the curve y = x^2 will be calculated at x = 2.
The distance is given by x - 2 = 2 - 2 = 0.
So, the radius of each cylindrical shell is 0.
Step 4: Find the volume of each cylindrical shell
The volume of each cylindrical shell is given by the formula V = 2πrh, where r is the radius and h is the height.
Substituting the values from the previous steps, we have V = 2π(0)(4) = 0.
Since the radius of each cylindrical shell is 0, the volume of each shell is 0.
Step 5: Sum up the volumes of all the cylindrical shells
Since the volume of each cylindrical shell is 0, the total volume of the solid formed by revolving region R about the line x = 2 is also 0.
Therefore, the volume of the solid is 0 cubic units.
To find the volume of the solid formed by revolving R about the line x = 2, we can use the method of cylindrical shells.
First, let's visualize the given region R. The x-axis, x = 2, and the curve y = x^2 form a bounded region:
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Now, to construct cylindrical shells, we imagine slicing R into thin vertical strips parallel to the y-axis. Each strip has a width of Δy, and its height is given by the difference between the x-coordinate of the curve y = x^2 and x = 2.
Let's consider a strip at a particular y-coordinate (y-value). As the strip revolves about the line x = 2, it forms a cylindrical shell. The height of this shell is 2 units (distance from x = 2 to the y-axis), and its radius is given by subtracting the x-coordinate of the curve y = x^2 from 2.
The volume of each cylindrical shell can be calculated as the product of its height, circumference, and thickness (Δy):
V_shell = 2π * (2 - x^2) * Δy
To find the total volume of the solid, we need to sum up the volumes of all these cylindrical shells. Mathematically, we integrate the expression for the volume with respect to y:
V_total = integral[R] (2π * (2 - x^2)) dy
To proceed, we need to express the curve y = x^2 in terms of y to determine the limits of integration.
Rearranging the equation, we get:
y = x^2 ---> x^2 = y ---> x = √y
Now, let's determine the limits of integration. As y increases, x starts from the x-axis (x = 0) and reaches x = 2 when it intersects with the vertical line x = 2. Therefore, the limits of integration for y are from 0 to 4 since x = √y for y between 0 and 4.
Now that we have the setup, we can evaluate the integral:
V_total = integral[0 to 4] (2π * (2 - x^2)) dy
Now solve the integral and calculate the volume to obtain the final answer.