Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
y = x^2, y = 0, x = 0; x = 3, about the y-axis
2007-07-28 18:16:38 · 3 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
V = ∫ π x² dy between 0 and 9
V = ∫ π y dy lims 0 to 9
V = π [ y ² / 2 ] lims 0 to 9
V = 81 π / 2
2007-07-28 20:05:49 · answer #1 · answered by Como 7 · 1⤊ 0⤋
Use disk method.
Find the intersection of the curves to determine bounds. When x = 0, y = 0. When x = 3, y = 9. The outside radius is 3. The inside radius is sqrt(y). So the integral we're looking for is:
pi * Int (0 to 9) [3^2 - ((sqrt(y))^2]dy
= pi * [9y - (1/2)y^2] evaluated at (0 to 9)
= pi * [81 - (1/2)*81]
= (81/2)*pi
2007-07-29 02:28:42 · answer #2 · answered by pki15 4 · 0⤊ 0⤋
Volume of delta cylinder along y axis by limiting y to 0 is
dV = pi.x^2.dy
Since y=x^2,
dV = pi.y.dy
Volume = Integral of dV
= pi.(y^2)/2
= pi.(x^4)/2
For 0 ⤠x ⤠3,
Volume = pi.81/2 = 40.5 pi
2007-07-29 02:40:53 · answer #3 · answered by miamidot 3 · 0⤊ 1⤋