Find the volume formed by rotating the region enclosed by: y=2sqrt(x) and y=x about the line y=4
The curves intersect at (0,0) and (4,4), so
using discs (washers)
v = ∫[0,4] π(R^2-r^2) dx
where R = 4-x and r = 4-2√x
∫[0,4] π((4-x)^2-(4-2√x)^2) dx = 32π/3
using shells,
v = ∫[0,4] 2πrh dy
where r=4-y and h=y-y^2/4
v = ∫[0,4] 2π(4-y)(y-y^2/4) dy = 32π/3
Why did the region cross the road? To find its volume after rotating about y=4, of course! Let's get calculating!
To find the volume formed by rotating the region enclosed by the curves y = 2√(x) and y = x about the line y = 4, we need to use the method of cylindrical shells.
The first step is to determine the limits of integration. This can be done by finding the x-values where the two curves intersect. Set the two equations equal to each other:
2√(x) = x
Square both sides:
4x = x^2
Rearrange the equation:
x^2 - 4x = 0
Factor out x:
x(x - 4) = 0
Solve for x:
x = 0 or x = 4
So the limits of integration will be from x = 0 to x = 4.
Next, we need to find the height of each cylindrical shell. The height is given by the difference between y = 4 and the y-values of the two curves at each x-value.
At x = 0, the height of the cylindrical shell is y = 4 - 2√(0) = 4.
At x = 4, the height of the cylindrical shell is y = 4 - 2√(4) = 4 - 4 = 0.
Now, we need to find the radius of each cylindrical shell. The radius is given by the distance from the axis of rotation (y = 4) to each curve at each x-value.
At x = 0, the radius of the cylindrical shell is 4 - 0 = 4.
At x = 4, the radius of the cylindrical shell is 4 - 4 = 0.
Now we can set up the integral to find the volume:
V = ∫[from x = 0 to x = 4] 2πr * h dx
V = ∫[from x = 0 to x = 4] 2π(4)(4 - x) dx
V = 2π ∫[from x = 0 to x = 4] (16 - 2πx) dx
V = 2π(16x - (πx^2)/2) |[from x = 0 to x = 4]
V = 2π((16(4) - (π(4)^2)/2) - (16(0) - (π(0)^2)/2))
V = 2π(64 - 32π/2 - 0)
V = 2π(64 - 16π)
V = 128π - 32π^2
Therefore, the volume formed by rotating the region enclosed by y = 2√(x) and y = x about the line y = 4 is 128π - 32π^2 cubic units.
To find the volume formed by rotating a region, we can use the method of cylindrical shells.
First, let's sketch the given region and the line of rotation:
Region: y = 2√x
Line of rotation: y = 4
Graphing these two equations, we can see that they intersect at x = 4.
To determine the limits of integration, we need to find the x-values where the region is bounded. In this case, the region is bounded by x = 0 and x = 4, as shown on the graph.
Now, let's consider a small vertical strip within the region, at a distance x from the y-axis. The width of this strip is dx.
The height of the strip is given by the difference between the two functions: y = 4 - (2√x - x)
To calculate the volume of this strip, we multiply its height by the circumference of the resulting cylindrical shell. The circumference of a cylindrical shell is 2πr, where r is the distance from the strip to the line of rotation.
The distance from the strip to the line y = 4 is just the difference between 4 and the height of the strip.
So, the volume of this small cylindrical shell is approximately: dV = 2π(4 - (2√x - x)) dx
To find the total volume, we integrate this expression with respect to x over the interval [0, 4]:
V = ∫[0,4] 2π(4 - (2√x - x)) dx
This integral will give us the volume formed by rotating the region enclosed by y = 2√x and y = x about the line y = 4.