The volume of the solid formed by rotating once about the x-axis the region bounded by the curve y = f(x), the x-axis, x = a, and x = b is
V = π·∫_a^b {f(x)}² dx
Example ① The volume of the solid formed by rotating the region bounded by y = √x, the x-axis, x=0, and x=4 about the x-axis.
Solution: V = π∫_0^4 (√x)² dx = π∫_0^4 x dx = π·[x²/2]_0^4 = 8π
Solution: V = π∫_0^4 (√x)² dx = π∫_0^4 x dx = π·[x²/2]_0^4 = 8π
Example ② (Sphere) The volume of a sphere of radius r.
Solution: Rotate y = √(r²−x²) about the x-axis.
V = π∫_{−r}^{r} (r²−x²) dx = π[r²x − x³/3]_{−r}^{r} = (4/3)πr³
Solution: Rotate y = √(r²−x²) about the x-axis.
V = π∫_{−r}^{r} (r²−x²) dx = π[r²x − x³/3]_{−r}^{r} = (4/3)πr³
Key Point
- About the x-axis: V = π∫ y² dx
- About the y-axis: V = π∫ x² dy (integrate with respect to y)
Practice Problems
- The volume obtained by rotating y = x², the x-axis, x=0, and x=1 about the x-axis.
Solution and Explanation
- Answer: π/5
π∫_0^1 x⁴ dx = π/5。