Question

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. y=−3x+6, y=3x, x= 0

Answer 1

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.