The region bounded by y = x², x = 0, and x = 3 is revolved about the y-axis. Which integral correctly applies the shell method?
A∫₀⁹ π(√y)² dy — integrating disks in the y-direction
B∫₀³ 2π · x · x² dx — shells with radius x, height x², thickness dx
C∫₀³ π · (x²)² dx — integrating disks in the x-direction
D∫₀³ 2π · x² · x dx — shells with radius x², height x
The shell method for revolution about the y-axis gives V = ∫ 2π · (radius) · (height) dx. Each vertical strip at position x has radius x (distance from the y-axis) and height f(x) = x². So the integral is ∫₀³ 2πx · x² dx = ∫₀³ 2πx³ dx. Option A is the washer method (requiring the inverse function √y), which is correct but integrates in y. Option C uses π instead of 2π — the disk formula. Option D swaps radius and height.
Question 2 Multiple Choice
For which setup does the shell method offer the clearest advantage over the washer/disk method?
ARevolving about the x-axis when the function is already given as y = f(x)
BRevolving about the y-axis when f(x) is given explicitly but its inverse f⁻¹(y) would be difficult to compute
CWhen the region has a hole, since shells handle washers better than the washer method
DWhen the function is linear, because the shells have constant height
The shell method's chief advantage is that it lets you integrate in the same variable as the function is expressed. If y = f(x) and you revolve about the y-axis, the washer method requires x = f⁻¹(y), which may be impossible in closed form. The shell method integrates in x directly: ∫ 2πx·f(x)dx. Option A is backwards — revolving about the x-axis with y = f(x) is where the washer method is natural. Option C confuses having a hole with the choice of method.
Question 3 True / False
The shell method and the washer method both use the factor π in their volume element formulas.
TTrue
FFalse
Answer: False
The washer method uses π because it computes the area of a circular cross-section (πr²). The shell method uses 2π because it computes the circumference of a cylindrical shell (2πr) and multiplies by height and thickness. The 2π comes from unwrapping the shell into a flat slab: volume ≈ circumference × height × thickness = 2πr · h · dr. Forgetting this distinction and using π in the shell formula is a very common error.
Question 4 True / False
When revolving a region about the line x = 4 instead of the y-axis, the radius of each shell at position x is simply x.
TTrue
FFalse
Answer: False
When the axis of revolution is not at the origin, the radius must measure the distance from each strip to that axis. For revolution about x = 4, a strip at position x has radius |x − 4|, which equals 4 − x when x < 4 and x − 4 when x > 4. Using just x instead of |x − 4| is a common error when the axis is shifted. The formula remains 2π · (radius) · (height) · dx, but the radius expression changes.
Question 5 Short Answer
Explain why the shell method avoids the need to find an inverse function when revolving the region under y = f(x) about the y-axis, and what each factor in the integrand 2π · x · f(x) dx represents geometrically.
Think about your answer, then reveal below.
Model answer: The shell method decomposes the solid into thin cylindrical shells parallel to the y-axis. Each vertical strip at position x (with width dx) is revolved around the y-axis to form a shell. Its radius is x (distance from the y-axis), its height is f(x) (the function value), and its wall thickness is dx. Unwrapping the shell into a flat slab gives volume ≈ 2πx · f(x) · dx — circumference times height times thickness. Integrating over x from a to b gives the total volume without ever needing to express x in terms of y. The washer method slices perpendicular to the y-axis, requiring y as the integration variable and x = f⁻¹(y) as the horizontal extent — the inverse function.
Shells are parallel to the axis of revolution; washers are perpendicular to it. Parallel shells let you integrate in the same variable as the function, sidestepping the inversion problem.