Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis. x = 7y^2, y ≥ 0, x = 7; about y = 2
the curves intersect at (7,1) and (7,-1)
Using shells of thickness dy, we have
v = ∫[-1,1] 2πrh dy
where r = 2-y and h = 7-7y^2
v = ∫[-1,1] 14π(2-y)(1-y^2) dy = 112π/3
Using discs (washers) of thickness dx, we have
v = ∫[0,7] π(R^2-r^2) dx
where R = 2+√(x/7) and r = 2-√(x/7)
v = ∫[0,7] π((2+√(x/7))^2-(2-√(x/7))^2) dx = 112π/3
To find the volume generated by rotating the region bounded by the curves x = 7y^2, y ≥ 0, x = 7, about the axis y = 2, we will use the method of cylindrical shells.
Step 1: Draw a rough sketch of the given curves and the axis of rotation.
Step 2: Determine the bounds of integration. Since y ≥ 0, the lower bound will be 0. To find the upper bound, we need to solve the equation x = 7 for y:
7y^2 = 7
y^2 = 1
y = ±1
However, since y ≥ 0, the upper bound will be 1.
Step 3: Determine the height of each cylindrical shell. The height of each shell is the difference between the axis of rotation (y = 2) and the curve given by x = 7y^2. So the height is 2 - 7y^2.
Step 4: Determine the radius of each cylindrical shell. The radius of each shell is the distance between the axis of rotation (y = 2) and the axis of symmetry (in this case, the y-axis). So the radius is 2.
Step 5: Write the integral expression for the volume V. The volume of each cylindrical shell is given by the formula V_shell = 2πrh, where r is the radius and h is the height. We need to integrate this expression over the interval [0, 1] to find the total volume:
V = ∫[0,1] (2πrh) dy
= ∫[0,1] (2π(2)(2 - 7y^2)) dy
= 4π ∫[0,1] (4 - 14y^2) dy
= 4π [4y - (14/3)y^3] |[0,1]
= 4π [(4(1) - (14/3)(1)^3) - (4(0) - (14/3)(0)^3)]
= 4π [(4 - 14/3) - (0 - 0)]
= 4π [(12/3 - 14/3)]
= 4π (-2/3)
= -8π/3
So the volume generated by rotating the region about the axis y = 2 is -8π/3 cubic units.
To find the volume generated by rotating the region bounded by the given curves about the axis y = 2 using the method of cylindrical shells, follow these steps:
Step 1: Visualize the region and axis of rotation:
Draw the graphs of the curves x = 7y^2 and x = 7 to understand the shape of the region. Also, draw the line y = 2 which represents the axis of rotation. Make sure to shade the region bounded by the curves.
Step 2: Determine the height of the cylindrical shells:
The height of each cylindrical shell will be the difference between the y-coordinate of the upper curve (x = 7) and the y-coordinate of the lower curve (x = 7y^2). Hence, the height of the cylindrical shell is given by h = 7 - 7y^2.
Step 3: Determine the radius of the cylindrical shells:
The radius of each cylindrical shell will be the distance between the axis of rotation (y = 2) and the x-coordinate of the lower curve (x = 7y^2). Hence, the radius of the cylindrical shell is given by r = 2 - y.
Step 4: Set up the integral:
The formula for the volume of a cylindrical shell is V = 2πrh, where r is the radius and h is the height.
Since we are rotating about the axis y = 2, the range of y-values will be given by the intersection points of the curves: 0 ≤ y ≤ √(7/2).
The volume V can be calculated by integrating the product of 2πrh over this range. Hence, the integral is:
V = ∫[0,√(7/2)] 2π(2 - y)(7 - 7y^2) dy.
Step 5: Solve the integral:
Evaluate the integral using standard integration techniques, such as u-substitution or expanding and simplifying the expression. This will give you the volume V of the region.
Note: To get the final numerical result, it is recommended to evaluate the integral using a calculator or computer algebra system.