※ 引述《william01 (小威)》之銘言：
: 請問各位大大 小弟有幾題的積分題目 我不會解 可以幫忙解一下嗎?
: Find the relative extrema of the function
: f(x,y) = 4y^3 + x^2 - 12y^2 - 36y + 2 ?
: Find the area of the region completely enclosed by the graphs
: of the functions
: f(x) = x^3 - 3x + 3 and g(x) = x + 3 ?
令 f(x) = g(x) , 則 x^3 - 3x + 3 = x + 3
x^3 - 4x = 0 => (x)(x^2 - 4) = 0 => (x)(x - 2)(x + 2) = 0
x = -2 , 0 , 2
畫圖可得知當 0 ≦ x ≦ 2 時 , x^3 - 4x ≦ x + 3
-2 ≦ x ≦ 0 時 , x^3 - 4x ≧ x + 3
0 2
因此所求面積 = ∫ (x^3 - 3x) - (x + 3) dx + ∫ (x + 3) - (x^3 - 3x) dx
-2 0
0 2
= ∫ x^3 - 4x dx + ∫ 4x - x^3 dx
-2 0
x^4 |0 x^4 |2
= --- - (2)(x^2) | + (2)(x^2) - --- |
4 |-2 4 |0
= 4 + 4 = 8
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