Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y^2 = 2x, x = 2y; about the y-axis
The curves intersect at (8,4), so using discs (washers) of thickness dy,
v = ∫ π(R^2-r^2) dy
where R=2y and r = y^2/2
v = ∫[0,4] π((2y)^2 - (y^2/2)^2) dy = 512π/15
Or, using shells of thickness dx,
v = ∫ 2πrh dx
where r=x and h = √(2x) - x/2
v = ∫[0,8] 2πx(√(2x) - x/2) dx = 512π/15
To find the volume V of the solid obtained by rotating the region bounded by the curves y^2 = 2x and x = 2y about the y-axis, we can use the method of cylindrical shells.
1. First, let's sketch the curves y^2 = 2x and x = 2y to visualize the region bounded by them. We can rewrite the equation y^2 = 2x as x = (1/2)y^2, which is a parabolic curve opening to the right.
2. To use cylindrical shells, we need to express y in terms of x. From the equation x = 2y, we can solve for y as y = (1/2)x.
3. Next, we need to determine the limits of integration. The region bounded by the curves will have a lower limit at x = 0 and an upper limit at the point of intersection of the two curves.
4. To find the point of intersection, we equate the equations y^2 = 2x and x = 2y:
(1/2)x = 2y
(1/2)x = 2(1/2)x^2
x = x^2
x^2 - x = 0
x(x - 1) = 0
Therefore, x = 0 or x = 1.
At x = 0, the corresponding value of y is y = (1/2)(0) = 0.
At x = 1, the corresponding value of y is y = (1/2)(1) = 1/2.
So, our limits of integration are from y = 0 to y = 1/2.
5. Now, let's set up the integral for calculating the volume using cylindrical shells. The volume of each cylindrical shell is given by the formula:
dV = 2πrh*dx,
where r is the radius from the y-axis to the shell (which is x), h is the height of the shell (which is the difference between the upper and lower y-values at a given x), and dx is the thickness of the shell.
Since the axis of rotation is the y-axis, the radius r is simply the x-value. The height h is the difference between the upper and lower y-values, which is y_upper - y_lower.
Therefore, the integral for the volume is:
V = ∫[0 to 1/2] 2πx * (y_upper - y_lower) * dx.
6. Now, substitute y_upper and y_lower into the equation. We already found that y_upper = 1/2x and y_lower = 0.
V = ∫[0 to 1/2] 2πx * ((1/2x) - 0) * dx
= π∫[0 to 1/2] x * dx.
7. Integrate the expression πx with respect to x over the interval [0, 1/2]:
V = π * [(1/2)(1/2)^2 - (1/2)(0)^2]
= π * (1/8).
8. Simplifying, we find that the volume of the solid is V = π/8 cubic units.
Therefore, the volume of the solid obtained by rotating the region bounded by the curves y^2 = 2x and x = 2y about the y-axis is π/8 cubic units.