Answer:
40π
Explanation:
First, find the limits (intersections).
5x² = 30x − 10x²
15x² − 30x = 0
x² − 2x = 0
x (x − 2) = 0
x = 0 or 2
Within this interval, 30x − 10x² is greater than 5x².
Dividing the volume into cylindrical shells, the volume of each shell is:
dV = 2π r h t
dV = 2π x (30x − 10x² − 5x²) dx
dV = 2π x (30x − 15x²) dx
dV = 30π (2x² − x³) dx
The total volume is the sum (integral):
V = ∫ dV
V = ∫₀² 30π (2x² − x³) dx
V = 30π ∫₀² (2x² − x³) dx
V = 30π (⅔ x³ − ¼ x⁴)|₀²
V = 30π (⅔ 8 − ¼ 16)
V = 30π (16/3 − 4)
V = 10π (16 − 12)
V = 40π
