The line 𝑦 = 2𝑥 from 𝑥 = 0 to 1 is rotated around the y-axis. The volume generated is:
A. 1/2 𝜋
B. 2/3 𝜋
C. 3/4 𝜋
D. 𝜋

 

 

Volume of Revolution: Line y = 2x Around the Y-Axis

If you’re studying calculus, one of the classic problems involves finding the volume of a solid formed by rotating a curve around an axis. Let’s walk through how to calculate the volume generated by rotating the line y = 2x from x = 0 to 1 around the y-axis.

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Step-by-Step Solution

We’re rotating around the y-axis, so we need to express x as a function of y:

Given: y = 2x ⇒ x = y/2

The volume V formed by rotating a function about the y-axis is calculated using the method of disks:

V = ∫[y₁ to y₂] π·(x)^2 dy

From x = 0 to 1, when y = 2x, the y-values go from:

y = 2×0 = 0 to y = 2×1 = 2

Now plug into the formula:

V = ∫[0 to 2] π·(y/2)^2 dy
  = π ∫[0 to 2] (y² / 4) dy
  = (π/4) ∫[0 to 2] y² dy
  = (π/4) × [y³ / 3] from 0 to 2
  = (π/4) × (8/3)
  = 2π/3

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✅ Final Answer: B. 2/3 𝜋

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Key Takeaways

• Always convert to x = f(y) when rotating around the y-axis.
• Use the disk method to integrate in terms of y.
• Double-check your bounds when switching variables.