Answer:
18π
Step-by-step explanation:
Graph: desmos.com/calculator/kpvlltsbhv
The region is defined by 3 lines, which form a triangle (the red area in the graph). Cut a thin vertical slice (the blue area in the graph) with a width of dx, at a position of x.
When we revolve this slice around the x-axis, we get a hollow cylinder (or a washer). The outside radius is -3x + 6, and the inside radius is 3x. So the volume of the cylinder is:
dV = π [(-3x + 6)² − (3x)²] dx
dV = π (9x² − 36x + 36 − 9x²) dx
dV = π (-36x + 36) dx
dV = -36π (x − 1) dx
Now let's find the limits. We know the beginning of the triangle is at x = 0. To find where the triangle ends, solve for the intersection of the other two lines:
-3x + 6 = 3x
6x = 6
x = 1
So the region is between x = 0 and x = 1.
That means the total volume is the sum of all the hollow cylinders' volumes between x = 0 to x = 1.
V = ∫₀¹ dV
V = ∫₀¹ -36π (x − 1) dx
Evaluating the integral:
V = -36π ∫₀¹ (x − 1) dx
V = -36π (½ x² − x) |₀¹
V = -36π [(½ 1² − 1) − (½ 0² − 0)]
V = -36π (½ − 1)
V = 18π
The volume of the solid is 18π cubic units.
