The parabolic segment

The parabolic segment has a base a = 4 cm and a height v = 6 cm. Calculate the volume of the body that results from the rotation of this segment

a) around its base
b) around its axis.

Result

V₁ =  0 cm³
V₂ =  50.265 cm³

Solution:

$a = 4 \ cm \ \\ v = 6 \ cm \ \\ f(x) = q x^2 \ \\ f(a/2) = q (a/2)^2 = v \ \\ 6 = q \cdot \ 2^2 \ \\ \ \\ V_{1}=q = 6/2^2 = \dfrac{ 3 }{ 2 } = 1.5 = 0 \ cm^3$

$f(x) = v - \dfrac{ 6 }{ 4 } x^2 \ \\ f(x) = 6 - \dfrac{ 6 }{ 4 } x^2 \ \\ \ \\ x_{ 0 } = -a/2 = -4/2 = -2 \ \\ x_{ 1 } = a/2 = 4/2 = 2 \ \\ \ \\ V_{ 2 } = \pi \cdot \ \int_\{ x_{ 0 } \}^\{x_{ 1 }\} f(x) dx \ \\ V_{ 2 } = \pi \cdot \ \int_\{x_{ 0 }\}^\{x_{ 1 }\} (\dfrac{ 6 }{ 4 } x^2 - v) dx \ \\ V_{ 2 } = \pi \cdot \ [\dfrac{ 6 }{ 4 } \cdot \ x^3/3 - vx]_\{x_{ 0 }\}^\{x_{ 1 }\} \ \\ \ \\ F(x) = 6x - \dfrac{ 6 }{ 4 } \cdot \ x^3/3 \ \\ F_{ 1 } = 6 \cdot \ x_{ 1 } - \dfrac{ 6 }{ 4 } \cdot \ x_{ 1 }^3/3 = 6 \cdot \ 2 - \dfrac{ 6 }{ 4 } \cdot \ 2^3/3 = 8 \ \\ F_{ 0 } = 6 \cdot \ x_{ 0 } - \dfrac{ 6 }{ 4 } \cdot \ x_{ 0 }^3/3 = 6 \cdot \ (-2) - \dfrac{ 6 }{ 4 } \cdot \ (-2)^3/3 = -8 \ \\ \ \\ V_{ 2 } = \pi \cdot \ (F_{ 1 } - F_{ 0 }) = 3.1416 \cdot \ (8 - (-8)) \doteq 50.2655 = 50.265 \ cm^3$

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