jack 發問時間: 科學數學 · 1 0 年前

use a triple integral to find

use a triple integral to find the volume of the given solid

The tetrahedron enclosed by the coordinate planes and

the plane 2x+y+z=4

2 個解答

評分
  • 1 0 年前
    最佳解答

    ∫∫∫ dzdydx (z from 0 to 4-2x-2y, y from 0 to 4-2x. x from 0 to 2)

    = ∫∫ 4-2x-y dydx

    = ∫ 4y-2xy-y^2/2 [0,4-2x] dx

    =1/2 ∫ (4-2x)^2 dx

    =1/2 (16x-8x^2+(4/3)x^3) [0,2]

    =16/3

  • linch
    Lv 7
    1 0 年前

    R = {(x, y,z) | 0 <= z <= 4 - 2x - y, 0 <= y <= 4 - 2x, 0 <= x <= 2}

    V =∫∫∫_R 1 dV

    =∫_[0, 2]∫_[0,4 - 2x]∫_[0, 4 - 2x - y] 1 dzdydx

    =∫_[0, 2]∫_[0,4 - 2x] z |_[0, 4 - 2x - y] dy dx

    =∫_[0, 2]∫_[0,4 - 2x] (4 - 2x - y) dy dx

    =∫_[0, 2] (4y - 2xy - y^2/2 ) |_[0, 4 - 2x] dx

    2009-06-10 21:18:23 補充:

    =∫_[0, 2] [4(4 - 2x) - 2x(4 - 2x) - (4 - 2x)^2/2 ] dx

    =∫_[0, 2] [ 2x^2 - 8x + 8 ] dx

    = 2x^3 / 3 - 4x^2 + 8x |_[0, 2]

    = 16/3 - 16 + 16

    = 16/3

還有問題?馬上發問,尋求解答。