Let and be real positive constants.
The volume of the region bounded by the curve , when revolved about the line , can be expressed as where and are coprime positive integers.
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The volume V = π ∗ ∫ 0 3 ∗ b ( x − a ) 2 d y =
π b 2 a 2 ∗ ∫ 0 3 ∗ b 2 b 2 + y 2 − 2 b b 2 + y 2 d y
Let I 2 = 2 b ∗ ∫ 0 3 ∗ b b 2 + y 2 d y ′
Using trig substitution, Let y = b t a n θ ⟹ d y = b s e c 2 θ ⟹
I 2 = 2 b 3 ∗ ∫ 0 3 π s e c 3 ( θ ) d θ
Using integration by parts let u = s e c θ ⟹ d u = s e c θ t a n θ
d v = s e c 2 ( θ ) ⟹ v = t a n θ ⟹
∫ 0 3 π s e c 3 ( θ ) d θ = s e c θ t a n θ ∣ 0 3 π − ∫ 0 3 π s e c 3 ( θ ) d θ + ∫ 0 3 π s e c θ d θ ⟹
2 ∗ ∫ 0 3 π s e c 3 ( θ ) d θ = ( s e c θ t a n θ + l n ∣ s e c θ + t a n θ ∣ ) ∣ 0 3 π ⟹
∫ 0 3 π s e c 3 ( θ ) d θ = 2 1 ∗ ( s e c θ t a n θ + l n ∣ s e c θ + t a n θ ∣ ) ∣ 0 3 π =
2 1 ∗ ( 2 3 + l n ( 2 + 3 ) ) ⟹
I 2 = b 3 ∗ ( 2 3 + l n ( 2 + 3 ) )
Let I 1 = ∫ 0 3 ∗ b 2 b 2 + y 2 d y ⟹
I 1 = 2 b 2 y + 3 y 3 ∣ 0 3 ∗ b = 3 3 b 3 ⟹
V = a 2 b π ∗ ( 3 − l n ( 2 + 3 ) ) = a 2 b π ∗ ( m − l n ( n + m ) ) ⟹
m + n = 5 .