Use any method to determine the volume of the solid of revolution described.

Question

y = -x +1 , y = x +1 , y = 0 , rotated about the line x = -2

Answer 1

Plotting the constraints, I find the area to be the triangle (-1,0), (0,1), (1,0). To revolve this around x = -2, we want to express each equation as x = -y+1 and x = y-1, use the washer method by adding 2 for the offset, and integrate in y from 0 to 1. Outer radius R = (-y+1) +2, inner r = (y-1) +2.

dV = (πR² - πr²) dy = π[(-y+3)² - (y+1)²] dy = π[y² -6y +9 -(y² +2y +1)] dy = π[-8y + 8] dy.

V = ∫dV = π [-4y² + 8y]₀¹ = 4π.