Find the volume of the solid formed by revolving the region bounded by f(x)= 2-x^2 and g(x)=1 about the line y=1.
To find the volume of the solid formed by revolving the region bounded by two curves around a line, we can use the method of cylindrical shells.
First, let's find the points of intersection between the curves f(x) and g(x).
Setting the two functions equal to each other:
2 - x^2 = 1
Rearranging the equation:
x^2 = 1
Taking the square root of both sides:
x = ±1
So the points of intersection are (1, 1) and (-1, 1).
Next, let's find the height of each cylindrical shell formed by revolving the region. The height of each shell is the difference between the y-coordinate of the upper curve and the y-coordinate of the lower curve. In this case, the height is:
h(x) = f(x) - g(x)
= (2 - x^2) - 1
= 1 - x^2
Now, let's find the radius of each cylindrical shell. The radius is the distance between the axis of revolution and the curve. In this case, the axis of revolution is the line y = 1, so the radius is given by:
r(x) = y - 1
= 1 - 1
= 0
Therefore, the radius of each cylindrical shell is 0.
The volume of each cylindrical shell is given by the formula:
V = 2πrh
Since the radius of each shell is 0, the volume can be simplified to:
V = 2πrh
= 2π(0)(1 - x^2)
= 0
Since the volume of each shell is 0, the total volume of the solid formed by revolving the region around the line y = 1 is also 0.
Therefore, the volume of the solid is 0.