Find the volume that is generated by rotating the area bounded by y=x^2 and y=2-x^2 about the x-axis using the shell method.
so, the shells are of thickness dy, and since the region is symmetric about the y-axis, the volume is
2∫[0,1] 2πrh dy
where r=y and h changes from √y to √(2-y) at y=1
v = 2∫[0,1] 2πy√y dy + 2∫[1,2] 2πy√(2-y) dy = 16π/3
check, using discs of thickness dx,
v = 2∫[0,1] π(R^2-r^2) dx
where R = 2-x^2 and r = x^2
v = 2∫[0,1] π((2-x^2)^2 - (x^2)^2) dx = 16π/3
To find the volume generated by rotating the area bounded by the curves y = x^2 and y = 2 - x^2 about the x-axis using the shell method, we can follow these steps:
Step 1: Determine the region of integration.
To find the bounds of integration, we need to find the points where the two curves intersect. Set x^2 = 2 - x^2 and solve for x:
2x^2 = 2
x^2 = 1
x = ±1
So the region of integration is bounded by x = -1 and x = 1.
Step 2: Set up the integral.
The general formula for the volume using the shell method is given by:
V = 2π∫[a,b] x * h(x) dx,
where a and b represent the bounds of integration, x is the variable of integration, and h(x) denotes the height of the shell at a given value of x. In this case, h(x) corresponds to the vertical distance between the curves y = x^2 and y = 2 - x^2.
Step 3: Express h(x) in terms of x.
To determine the height of the shell h(x), we subtract the lower curve (y = x^2) from the upper curve (y = 2 - x^2):
h(x) = (2 - x^2) - x^2
h(x) = 2 - 2x^2.
Step 4: Evaluate the integral.
The volume V can now be expressed as:
V = 2π∫[-1,1] x * h(x) dx
V = 2π∫[-1,1] x * (2 - 2x^2) dx.
Integrating this expression will yield the volume generated by the rotation.
Note: The shell method is a technique commonly used to find the volume of solids of revolution. It involves integrating the product of the circumference of a shell (2πx) and its height (h(x)) over the region of integration.