curves y^2=x and y=x-2 are rotated 360 degree over line x=4. What is the volume of the obtained solid?

Find points of intersection of the curves.

y = y² - 2

0 = y² - y - 2

0 = (y - 2)(y + 1)

y = -1, 2

(-1)² = 1 --> (1, -1)

2² = 4 --> (4, 2)

Finding volume using infinitesimal washers.

∫ π[(ro)² - (ri)²] dh =

∫ π[(4 - xo)² - (4 - xi)²] dy (over -1 ≤ y ≤ 2) =

∫ π[(4 - y²)² - (4 - (y + 2))²] dy =

∫ π[(4 - y²)² - (2 - y)²] dy =

∫ π[(16 - 8y² + y⁴) - (4 - 4y + y²)] dy =

∫ π[12 + 4y - 9y² + y⁴] dy =

π[12y + 2y² - 3y³ + y⁵/5] =

π[12(2 - (-1)) + 2(2² - (-1)²) - 3(2³ - (-1)³) + (2⁵ - (-1)⁵)/5] =

π[36 + 6 - 27 + 33/5] =

π[15 + 33/5] =

108π/5