If the curves of f(x) and g(x) intersect x=a and x=b and if f(x)>g(x)>0 for all x on (a,b) then the volume obtained when the region bounded by the curves is rotated about the x-axis is equal to
using discs,
v = = ∫[a,b] π(R62-r^2) dx
where R=f(x) and r=g(x)
using shells, it gets a lot trickier, because we have to idea of the behavior of f and g in the interval.
To find the volume of the solid obtained when the region bounded by the curves f(x) and g(x) is rotated about the x-axis, we can use the method of cylindrical shells.
The volume of a single cylindrical shell is given by the formula V = 2πrhΔx, where r is the distance from the axis of rotation to the shell, h is the height of the shell, and Δx is the small width of the shell.
In this case, the distance from the axis of rotation to the shell is the value of x itself. So, we have r = x.
The height of the shell, h, can be found by subtracting the y-values of the curves at the given x-value. In this case, h = f(x) - g(x).
The width of the shell, Δx, can be thought of as an infinitesimally small change in x, which we can represent as dx. So, we have Δx = dx.
To find the volume of the entire solid, we need to find the sum of the volumes of all the cylindrical shells. This requires integrating the expression for the volume of a single shell over the interval (a, b).
∫[a,b] 2πrh dx = ∫[a,b] 2πx(f(x) - g(x)) dx.
Therefore, the volume obtained when the region bounded by the curves is rotated about the x-axis is equal to ∫[a,b] 2πx(f(x) - g(x)) dx.
To find the volume obtained when the region bounded by the curves is rotated about the x-axis, we can use the method of cylindrical shells.
The volume of a cylindrical shell is given by the formula:
V = 2π ∫ [radius * height * thickness]
In this case, the radius of the cylindrical shell is the distance from the x-axis to the curve, the height is the difference between the values of f(x) and g(x) at a given x, and the thickness is an infinitesimally small change in x.
First, let's find the equation of the curves. We know that the curves intersect at x=a and x=b, so we can write the equations of the curves as f(x) and g(x):
f(x) = y = ...
g(x) = y = ...
Next, let's find the bounds of integration. The region bounded by the curves is defined by x ∈ [a, b]. Therefore, our integral will be taken over this interval.
Now, let's find the difference in the values of f(x) and g(x) for a given x on the interval (a, b). This will be our height:
Height = f(x) - g(x).
Finally, we need to determine the radius of the cylindrical shell. The radius is the distance from the x-axis to the curve. Since we are rotating the region around the x-axis, the radius is simply the x-coordinate itself, which is:
Radius = x.
Now we have all the necessary components to set up the integral for the volume:
V = 2π ∫ [Height * Radius * dx] for x ∈ [a, b].
Integrate the expression with respect to x over the given interval [a, b] to obtain the volume.