Let f(x)=x^2 and g(x)=4x-x^2. If R is the region bounded by f(x) and g(x), find the volumes of the solids generated when R is rotated about the line y=-2 and about the line x=-2.

Answer 1

1) Rotated about y=-2

Use the washer method

Limits: x² = 4x-x²

2x²-4x = 0 --> x = 0, 2

V = π ∫₀²(g(x) -d)² - (f(x) -d)²dx ,, d = -2

V= π ∫₀²(4x-x²+2)²-(x²+2)dx

= π ∫₀² -8x³ +8x²+16x dx

= π[-2x⁴+8x³/3+8x²]₀² = π(-32+64/3+32)

= 64π/3

2) Rotated about x = -2

Use shell method

V = 2π ∫₀² (x-d)(g(x)-f(x))dx

= 2π ∫₀² (x+2)(4x-x²-x²)dx = 2π ∫₀² (x+2)(4x-2x²) dx = 2π ∫₀² -2x³+8x dx

= 2π[-x⁴/2 + 4x²]₀² = 2π(8) = 16π