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11 May, 04:08

Electric charge is distributed over the rectangle 2 ≤ x ≤ 4, 0 ≤ y ≤ 2 so that the charge density at (x, y) is σ (x, y) = 2xy + y2 (measured in coulombs per square meter). Find the total charge on the rectangle.

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Answers (2)
  1. 11 May, 07:26
    0
    The total charge on the rectangle is 29.33C

    Step-by-step explanation:

    Given

    σ (x, y) = 2xy + y²

    For 2 ≤ x ≤ 4, 0 ≤ y ≤ 2

    The total charge on the rectangle is calculated as follows.

    Charge, q = ∫∫2xy + y² dydx {0,2}{2,4}

    Integrate with respect to y

    q = ∫xy² + ⅓y³ dydx {0,2}{2,4}

    q = ∫ (2²*x + ⅓*2³) dx {2,4}

    q = ∫4x + 8/3 dx {2,4}

    Integrate with respect to x

    q = 2x² + 8x/3 {2,4}

    q = (2 (4) ² + 8 (4) / 3) - (2 (2) ² + 8 (2) / 3)

    q = 29.33C
  2. 11 May, 07:54
    0
    Answer: The answer is 29.33C

    Step-by-step explanation:

    From the question above, we have the following:

    σ (x, y) = 2xy + y²

    For 2 ≤ x ≤ 4, 0 ≤ y ≤ 2

    The total charge on the rectangle will be calculated as follows.

    Charge, q = ∫∫2xy + y² dydx {0,2}{2,4}

    Now, we carry out integration with respect to y as follows:

    q = ∫xy² + ⅓y³ dydx {0,2}{2,4}

    q = ∫ (2²*x + ⅓*2³) dx {2,4}

    q = ∫4x + 8/3 dx {2,4}

    Next, we carry out integration with respect to x as follows:

    q = 2x² + 8x/3 {2,4}

    q = (2 (4) ² + 8 (4) / 3) - (2 (2) ² + 8 (2) / 3)

    q = (2 (16) + 32/3) - (2 (4) + 16/3)

    q = (32 + 10.67) - (8 + 5.33)

    q = 42.67 - 13.33

    q = 29.34C

    Therefore, the total charge on the rectangle is 29.34C
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