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 = 4x − x2, y = 3; about x = 1
each shell has thickness dx, so
v = ∫[1,3] 2πrh dx
where r=x-1 and h=y-3
v = ∫[1,3] 2π(x-1)(4x-x^2-3) dx = 8π/3
as a check, using discs of thickness dy,
v = ∫[3,4] π(R^2-r^2) dy
where R=(2+√(4-y)-1) and r=(2-√(4-y)-1)
v = ∫[3,4] π((1+√(4-y))^2-(1-√(4-y))^2) dy = 8π/3
To find the volume generated by rotating the region bounded by the given curves about the specified axis using the method of cylindrical shells, follow these steps:
Step 1: Draw a sketch of the given curves and the rotation axis.
Start by drawing the two curves: y = 4x - x^2 and y = 3. Then, draw the line x = 1, which represents the rotation axis. Make sure your sketch is accurate and clear.
Step 2: Determine the height of each cylindrical shell.
The height of each cylindrical shell is the difference in y-values between the two curves at a given x-value. In this case, the height can be found by subtracting the equation for the lower curve, y = 4x - x^2, from the equation for the upper curve, y = 3.
Height = 3 - (4x - x^2)
= x^2 - 4x + 3
Step 3: Determine the radius of each cylindrical shell.
The radius of each cylindrical shell is the distance between the x-value of the rotation axis and the x-value of a point on the curves. Since the rotation axis is x = 1, the distance can be found by subtracting 1 from the x-value.
Radius = x - 1
Step 4: Determine the differential element of volume.
The differential element of volume, dV, for each cylindrical shell is given by:
dV = 2πrh*dx
where r is the radius and h is the height of the cylindrical shell, and dx represents an infinitesimally small change in x.
Step 5: Integrate to find the total volume.
To find the total volume, integrate the differential element of volume over the range of x-values that covers the bounded region. In this case, you'll integrate from the x-values where the curves intersect.
V = ∫(from x1 to x2) 2πrh*dx
= ∫(from x1 to x2) 2π(x-1)(x^2-4x+3) dx
Evaluate the integral to get the value of the volume V. Make sure to substitute the appropriate limits of integration (x1 and x2) based on the points where the curves intersect.