Re: [積分] 求體積
※ 引述《pova (請叫我去睡覺)》之銘言：
: Find the Volume of the solid formed by revolving the region bounded by
: the graphs of f(x)=-x^2 +2x+3 and the x-axis about the x-axis
: ---------------------------------------------------------------
: 感謝
令 f(x) = -x^2 + 2x + 3 = 0
則 x^2 - 2x - 3 = 0 => (x - 3)(x + 1) = 0 => x = 3 , -1
所以 f(x) = -x^2 + 2x + 3 與 x 軸的交點為 (3,0) , (-1,0)
3
V = ∫ (π)*((f(x))^2) dx
-1
3
= ∫ (π)*((-x^2 + 2x + 3)^2) dx
-1
3
= (π)*(∫ x^4 + 4x^2 + 9 - 4x^3 + 12x - 6x^2 dx)
-1
3
= (π)*(∫ x^4 - 4x^3 - 2x^2 + 12x + 9 dx)
-1
x^5 2x^3 |3
= (π)*(----- - x^4 - ------ + 6x^2 + 9x) |
5 3 |-1
243 -1 2
= (π)*((--- - 81 - 18 + 54 + 27) - ((---) - 1 + --- + 6 - 9))
5 5 3
244 2
= (π)*(--- - 14 - ---)
5 3
732 - 210 - 10
= (π)*(---------------)
15
512
= (π)*(-----)
15
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