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Anonymous asked in 科學數學 · 1 decade ago


"Estimate the area between x = 0 and x = 1

under the graph of the function f(x) = x3 using

a. a lower sum with two rectangles of equal


b. an upper sum with two rectangles

of equal width.

c. the midpoint rule with two

rectangles of equal width. "




1 Answer

  • Leslie
    Lv 7
    1 decade ago
    Favorite Answer

    我們要拿 x 軸當做左邊及右邊, 共兩個長方形的底邊,

    然後各取它們的高度, 最後, 把兩個長方形的面積加起來, 就是答案.


    x = 0 到 x = 1, 長度為 1, 分割為兩段.

    左段及右段各當為左邊及右邊長方形的底 (寬度 = 0.5),

    至於左邊及右邊長方形的高(或叫做 "長"), 則取為:

    題 (a)--拿左段及右段的左端點的高度 (f(0) 及 f(0.5)) 當長,

    題 (b)--拿左段及右段的右端點的高度 (f(0.5) 及 f(1)) 當長,

    題 (c)--拿左段及右段的中點的高度 (f(0.5) 及 f(0.75)) 當長,

    把兩個長方形的面積 (長 * 寬) 加起來, 就是答案:

    a. f(0)*(0.5) 加 f(0.5)*(0.5)

    = 0 加 ((0.5)^3)*0.5

    = (0.5)^4

    b. f(0.5)*(0.5) 加 f(1)*(0.5)

    = ((0.5)^3)*0.5 加 1*(0.5)

    = (0.5)^4 加 0.5

    c. f(0.25)*(0.5) 加 f(0.75)*(0.5)

    = ((0.25)^3)*(0.5) 加 ((0.75)^3)*(0.5)

    (這是用 Riemann Sum 求 x^3 在 [0, 1] 間的積分 的近似值; 此題近似效果很差, 因為只分割成兩份; 分割越多直覺上效果越好.)

    Source(s): CALCULUS textbook
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