Anonymous
Anonymous asked in 科學數學 · 1 decade ago

(數學)微積分的問題!! (積分)

Find the areas of the regions enclosed by the following curves

a. x+y2 = 3 and 4x+y2 = 0 .

b. y=2sinx and y=sin2x, 0<= x <= π

請幫幫我這兩個問題!!

謝謝!!!

1 Answer

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  • Anonymous
    1 decade ago
    Favorite Answer

    (a) Points of intersection between the curves:

    x+y² = 3 and 4x+y² = 0

    y² = 3 - x = -4x

    3 = -3x

    x = -1

    y² = 4

    y = ±2

    Therefore the points of intersection are (-1, 2) and (-1, -2)

    So the area can be found as:

    A = ∫(-2 → 2) (x_1 - x_2)dy

    = ∫(-2 → 2) [(3 - y²) - (-y²/4)] dy

    = ∫(-2 → 2) (3 - 3y²/4) dy

    = [3y - y³/4] (-2 → 2)

    = 8 sq. units

    (b) Points of intersection between the curves:

    y = 2 sin x and y = sin 2x, 0<= x <= π

    2 sin x = sin 2x

    2 sin x = 2 sin x cos x

    sin x (1 - cos x) = 0

    x = 0 or π

    So the points of intersection are (0, 0) and (π, 0).

    So the area can be found as:

    A = ∫(0 → π) (y_1 - y_2)dx

    = ∫(0 → π) (2 sin x - sin 2x)dx

    = [-2 cos x + 0.5 cos 2x] (0 → π)

    = [2 cos x - 0.5 cos 2x] (π → 0)

    = 4 sq. units

    Source(s): My Maths knowledge
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