Answer:
38π cubic units
Step-by-step explanation:
The volume V = ∫₀²πy²dx
Since y = 3x + 1, dy = 3dx ⇒ dx = dy/3
So, when x = 0, y = 3(0) + 1 = 0 + 1 = 1
when x = 2, y = 3(2) + 1 = 6 + 1 = 7
So, changing the variable to y and the limits of integration from y = 1 to y = 7, we have
V = ∫₀²πy²dx
V = π∫₁⁷y²dy/3
V = π[y³/3]₁⁷/3
V = π[y³]₁⁷/9
V = [7³ - 1³]π/9
V = [343 - 1]π/9
V = 342π/9
V = 38π cubic units
