The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.
x = (y − 7)^2, x = 16; about y = 3
using shells of thickness dy,
v = ∫[3,11] 2πrh dy
where r = y-3 and h = 16-(y-7)^2
v = ∫[3,11] 2π(y-3)(16-(y-7)^2) dy = 2048π/3
using discs (washers) of thickness dx,
v = ∫[0,16] π(R^2-r^2) dx
where r = 7-√x-3 and R = 7+√x-3
v = ∫[0,16] π((4+√x)^2-(4-√x)^2) dx = 2048π/3
To find the volume of the resulting solid, we can use the method of cylindrical shells.
First, let's sketch the region bounded by the given curves:
The curve x = (y - 7)^2 is a parabola that opens to the right and intersects the x-axis at (49, 0). The line x = 16 is a vertical line. The region bounded by these curves forms a region in the first quadrant.
Next, let's identify the limits of integration. We need to integrate along the y-axis from the lowest y-value to the highest y-value of the region. From the equation x = (y - 7)^2, we can solve for y as follows:
x = (y - 7)^2
√x = y - 7
y = √x + 7
The lowest y-value is when x = 0:
y_min = √0 + 7 = 7
The highest y-value is when x = 16. Therefore, y_max = √16 + 7 = 11.
Now, we can set up the integral to find the volume V:
V = ∫[y_min, y_max] 2πy * (x_max - x_min) dy
First, let's find the expression for (x_max - x_min):
The line x = 16 does not affect the width of the region as we rotate it about the y-axis. Therefore, (x_max - x_min) = 0 - (y - 7)^2 = - (y - 7)^2.
Now, we can write the integral for the volume:
V = ∫[7, 11] 2πy * (-(y - 7)^2) dy
Simplifying the integral, we have:
V = -2π ∫[7, 11] y * (y - 7)^2 dy
Now, we can integrate this expression to find the volume V.
To find the volume of the resulting solid when the region bounded by the curves is rotated about the specified axis, we can use the method of cylindrical shells.
First, let's sketch the region bounded by the curves:
The curve x = (y - 7)^2 is a parabola shifted 7 units to the right. The curve x = 16 is a vertical line. The region bounded by these curves is between y = 3 and y = 16.
Next, we'll set up the integral to calculate the volume using cylindrical shells. The volume of a cylindrical shell is given by the formula:
V = ∫(2π * r * h) * dx
Where r is the distance from the axis of rotation to the shell (which in this case is y - 3), and h is the height of the shell (which is the difference between the x-coordinates of the curves at a given y-value).
To find the limits of integration, we need to determine the y-values where the curves intersect. Setting the two equations equal to each other:
(y - 7)^2 = 16
Taking the square root of both sides:
y - 7 = ±√(16)
y - 7 = ±4
y = 11 or y = 3
Since we're rotating about the line y = 3, the limits of integration are y = 3 to y = 11.
Now, we can express the integral in terms of y:
V = ∫(2π * (y - 3) * (16 - (y - 7)^2)) * dy
Integrating this expression over the limits y = 3 to y = 11 will give us the volume of the resulting solid.
Note: The intermediate steps involving the integral calculation are omitted here, as they can be quite involved.