We use integration to find the volume of the solid formed by rotating the curve y = f(x) around the x-axis.
Volume Formula for a Solid of Revolution
Solid of revolution around the x-axis: V = π ∫ₐᵇ {f(x)}² dx
Solid of revolution around the y-axis: V = π ∫_c^d {g(y)}² dy（using x = g(y)）
Example ①
Find the volume of the solid formed by rotating y = x²（0 ≤ x ≤ 2）around the x-axis.
Answer:V = π ∫₀² (x²)² dx = π ∫₀² x⁴ dx = π [x⁵/5]₀² = π × 32/5 = 32π/5
Find the volume of the solid formed by rotating y = x²（0 ≤ x ≤ 2）around the x-axis.
Answer:V = π ∫₀² (x²)² dx = π ∫₀² x⁴ dx = π [x⁵/5]₀² = π × 32/5 = 32π/5
Example ②
Find the volume formed by rotating y = √x（0 ≤ x ≤ 4）around the x-axis.
Answer:V = π ∫₀⁴ x dx = π [x²/2]₀⁴ = π × 8 = 8π
Find the volume formed by rotating y = √x（0 ≤ x ≤ 4）around the x-axis.
Answer:V = π ∫₀⁴ x dx = π [x²/2]₀⁴ = π × 8 = 8π
Example ③ (Washer Method)
Find the volume of the solid formed by rotating the region enclosed by y = x and y = x² around the x-axis.
Answer:Intersections x = 0, 1。On [0, 1], x ≥ x²。
V = π ∫₀¹ (x² − x⁴) dx = π [x³/3 − x⁵/5]₀¹ = π(1/3 − 1/5) = 2π/15
Find the volume of the solid formed by rotating the region enclosed by y = x and y = x² around the x-axis.
Answer:Intersections x = 0, 1。On [0, 1], x ≥ x²。
V = π ∫₀¹ (x² − x⁴) dx = π [x³/3 − x⁵/5]₀¹ = π(1/3 − 1/5) = 2π/15
Key Points
- About the x-axis: V = π∫{f(x)}²dx (the cross-section is a circle)
- Rotation of two curves: V = π∫({f}²−{g}²)dx (larger cross-section − smaller cross-section)
- Do not forget π
Practice Problems
- Find the volume formed by rotating y = 2x（0 ≤ x ≤ 3）around the x-axis.
- Find the volume formed by rotating y = x+1（0 ≤ x ≤ 2）around the x-axis.
- Find the volume formed by rotating the region enclosed by the graphs of y = x² and y = x around the x-axis.
Answers and Explanations
- Answer:36π
Explanation: V=π∫₀³4x²dx=π[4x³/3]₀³=π×36=36π。 - Answer:26π/3
Explanation: V=π∫₀²(x+1)²dx=π[(x+1)³/3]₀²=π(3³−1³)/3=π×26/3=26π/3。 - Answer:2π/15
Explanation: On [0,1], x≥x²。V=π∫₀¹(x²−x⁴)dx=π[x³/3−x⁵/5]₀¹=π(1/3−1/5)=2π/15。