Rotate x = g(y) about the y-axis → V = π∫_c^d {g(y)}² dy
Alternatively, use the “baumkuchen formula” (cylindrical shell method):
V = 2π·∫_a^b x·f(x) dx （when f(x) ≧ 0 on the interval [a,b]）
Example ① The volume obtained by rotating y = x², the x-axis, x=0, and x=2 about the y-axis.
Solution (baumkuchen formula): V = 2π∫_0^2 x·x² dx = 2π·[x⁴/4]_0^2 = 8π
Solution (baumkuchen formula): V = 2π∫_0^2 x·x² dx = 2π·[x⁴/4]_0^2 = 8π
Key Point
- For rotations about the y-axis, the baumkuchen formula is often convenient
Practice Problems
- The volume obtained by rotating y = √x, the x-axis, x=0, and x=1 about the y-axis.
Solution and Explanation
- Answer: 4π/5
V = 2π∫_0^1 x·√x dx = 2π·(2/5) = 4π/5。