it's easier to use y-axis as variable. The range is 0 to 5 for x,0 to 25 for y; y = x^2,x = √y
V = ∫₀²⁵π[5² - (√y)²]dy = π(25y - y²/2)|₀²⁵ = 625π/2
Just consider the first quadrant in a plane. A circle of raduius 15 is expressed as y = √(15² - x²); the hole can be considered as line y = 3; 3 = √(15² - x²),x² = 216
they inercept at (√216,3)
The volume is double the result from rotating the region about the x-axis
V = 2∫π[(15² - x²) - 3²]dx 0 to √216
= 2π(216x - x³/3) 0 to √216
= (1064√54)π/3
x = 0,y = 1; x = π/12,y = cos(π/2) = 0
The inner radius of the solid is r = 0 - (-6) = 6; the outer radius of the solid is R = cos(6x) - (-6) = 6 + cos(6x)
V = ∫π(((6 + cos(6x))² - 6²)dx 0 to π/12
= π∫[12cos(6x) + cos²(6x)]dx = π∫[12cos(6x) + cos²(6x)]dx
= π∫[12cos(6x) + 1/2 + (1/2)cos(12x)]dx
= π[x/2 + 2sin(6x) + (1/24)cos(12x)] 0 to π/12
= π(π/12 + 2 - 1/24) - π(0 + 0 + 1/24)
= π(2 + π/12)