Answer:
V = π ∫₀² (y² − 8y + 6√(2y)) dy
or
V = π ∫₀² (6x − 5x² + x³) dx
Explanation:
y₁ = 0.5x²
y₂ = x
First, find the intersections of the curves.
0.5x² = x
x² = 2x
x² − 2x = 0
x (x − 2) = 0
x = 0 or x = 2
So the points of intersection are (0, 0) and (2, 2).
When we revolve this region about the line x = 3, we get a hollow shape that looks like an upside-down funnel, or a volcano.
One option is to use washer method to find the volume, by cutting a thin horizontal slice of thickness dy, inner radius 3−x₁ = 3−√(2y), and outer radius of 3−x₂ = 3−y.
V = ∫₀² π [(3−y)² − (3−√(2y))²] dy
V = ∫₀² π (9 − 6y + y² − 9 + 6√(2y) − 2y) dy
V = π ∫₀² (y² − 8y + 6√(2y)) dy
Another option is to use shell method to find the volume, by cutting a thin vertical slice of thickness dx, radius 3−x, and height y₂−y₁ = x−0.5x².
V = ∫₀² 2π (3 − x) (x − 0.5x²) dx
V = ∫₀² 2π (3x − 1.5x² − x² + 0.5x³) dx
V = ∫₀² 2π (3x − 2.5x² + 0.5x³) dx
V = π ∫₀² (6x − 5x² + x³) dx
The second option is arguably easier to evaluate, but either one will get you the same answer (V = 8π/3).