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.
y=8-x^2, y= x^2 ; about x=2
V=?
the curves intersect where
8-x^2 = x^2
x = ±2
so, using shells,
v = ∫[-2,2] 2πrh dx
where r = 2-x and h = (8-x^2)-x^2
v = 2π∫(2-x)(8-2x^2) dx
= 256π/3
using discs (washers) and symmetry,
v = 2∫[0,2] π(R62-r^2) dy
where R=2+√y and r=2-√y
v = 2π∫[0,2] ((2+√y)^2 - (2-√y)^2) dy
To use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis, we need to follow these steps:
1. Sketch the region: On a graph, sketch the two curves y = 8 - x^2 and y = x^2. Make sure to include the axis of rotation, x = 2, in the sketch.
2. Determine the limits of integration: Find the x-values where the two curves intersect. Set the equations equal to each other and solve for x:
8 - x^2 = x^2
2x^2 = 8
x^2 = 4
x = ±2
Since we are rotating about the line x = 2, the limits of integration are x = -2 to x = 2.
3. Set up the integral: The formula for the volume of a cylindrical shell is given by:
V = ∫(2πrh)dx
Where:
- r is the distance from the axis of rotation to the shell (in this case, it is the distance from x = 2 to the curve)
- h is the height of the shell (in this case, it is the difference in y-values between the two curves)
- dx represents an infinitesimally small width in x
4. Calculate r and h: Due to the symmetry of the curves and the axis of rotation at x = 2, we only need to consider the positive values of x.
For a given x, the radius r is the difference between x and the axis of rotation (2):
r = x - 2
The height h is the difference in y-values between the two curves:
h = (8 - x^2) - x^2
h = 8 - 2x^2
5. Integrate the expression: Substitute the values of r and h into the volume formula and integrate with respect to x over the limits of integration (-2 to 2):
V = ∫(2π(x - 2)(8 - 2x^2))dx
6. Solve the integral: Perform the integration using the limits of integration:
V = ∫(2π(x - 2)(8 - 2x^2))dx from -2 to 2
7. Calculate the volume: Evaluate the integral to find the volume V.