he volume of the solid obtained by rotating the region bounded by x=(y−2)^2 and y=x about the x-axis has the form N/2π. What is the value of N?
To find the volume of the solid obtained by rotating the region bounded by x=(y−2)^2 and y=x about the x-axis, we can use the method of cylindrical shells.
First, let's sketch the graph of the given region to get a better understanding of the shape.
The equation x=(y−2)^2 represents a parabola opening to the right, with the vertex at (2, 0). The line y=x is a straight diagonal line passing through the origin (0,0). The region bounded by these curves is the area between the parabola and the line, as shown below.
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x-axis
To apply the method of cylindrical shells, we need to consider an infinitesimally thin vertical strip of height dx along the x-axis. Each strip will form a cylindrical shell when rotated about the x-axis. The radius of each shell is given by x, and the height is given by the difference between the upper and lower bounds of y.
To determine the limits of integration, we need to find the points of intersection between the parabola and the line. Setting y=(y−2)^2 equal to x, we have:
x = (y−2)^2
x = y^2 − 4y + 4
y^2 − 4y + 4 − x = 0
Using the quadratic formula, we can solve for y:
y = (4 ± sqrt(16 − 4(4−x)))/2
y = (4 ± sqrt(16 − 16 + 4x))/2
y = (4 ± sqrt(4x))/2
y = 2 ± sqrt(x)
From the graph, we can see that the region is bounded by y=2+sqrt(x) at the top and y=2-sqrt(x) at the bottom.
Now, let's integrate the volume element along the x-axis within the given bounds:
V = ∫[a,b] 2π*x*(h) dx
where a and b are the limits of integration, 2+sqrt(x) is the upper bound of y, 2-sqrt(x) is the lower bound of y, and h is the height of the cylindrical shell.
V = ∫[a,b] 2π*x * (2+sqrt(x) - (2-sqrt(x))) dx
V = ∫[a,b] 2π*x * (2*sqrt(x)) dx
V = ∫[a,b] 4π*sqrt(x) dx
To find the limits of integration, we need to find the x-values when the curves intersect. Setting 2+sqrt(x) = 2-sqrt(x), we have:
2+sqrt(x) = 2-sqrt(x)
2sqrt(x) = 0
sqrt(x) = 0
x = 0
Therefore, the bounds of integration are from x=0 to x=4.
V = ∫[0,4] 4π*sqrt(x) dx
Evaluating the definite integral, we find:
V = 4π * [(2/3) * x^(3/2)] from 0 to 4
V = 4π * [(2/3) * (4^(3/2) - 0^(3/2))]
V = (8π/3) * (2√2)
V = (16π√2) / 3
Thus, the volume of the solid obtained by rotating the region bounded by x=(y−2)^2 and y=x about the x-axis is (16π√2) / 3.
The value of N in the form N/2π is 16√2, so N = 16.