Mathématiques

Question

My document title: Calculus_1
Date yymmdd: 150206

Theme: AREAES

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Q1: Use RECTANGLES to find TWO approximations of the area of the region lying between the graph of { F(x) = -(x^2) + 4 } and the x-axis over [ 0; 2 ].

Q2: Find the upper and lower sums for the region bounded by the graph of { F(x) = x^2 } and the x-axis, between { x = 0 } and { x = 2}. (2)

Q3: Find the area of the region bounded by the graph of { F(y) = 4 - x^2 } , the x-axis , and { x = 2 }.

Q4: Find the area of the region bounded by the graph of { F(y) = y^2 } and the y-axis, for { 0 infequals y infequals 2 }.

Q5: Use the limit process to find the area of the region between the graph of the function and the x-axis over the given interval: { F(x) = (x^2) - x^3 } ; [ -1; +1 ].
My document title: Calculus_1 Date yymmdd: 150206 Theme: AREAES xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Q1: Use RECTANGLES to find TWO

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  • Q1:
             f(0) = 4           f(1) = 3
    On prend un rectangle de x = 0 a 1 de la hauteur f(0),   et    un rectangle de x = 1 a 2  de la hauteur f(1).
        Donc,  l’aire = 4 * 1 + 3 * 1 = 7

              f(0,5) = 3,75          f(1,5) =  1,75
    On prend  un rectangle entre  x = 0 et 1, de la hauteur f(0,5)    et   un rectangle entre x = 1  et 2 de la hauteur f(1,5).
             Donc  l’aire = 3,75 * 1 + 1,75 * 1 =  5,5
    =====================
    Q2:
             La limite inférieure:              f(0) * (2-0) = 0^2 * 2 = 0
                 La limite supérieure:              f(2) * (2 – 0) = 4 * 2 = 8
    =================
    Q3:
            x     ∈ [- 2, 2 ]
         soit      Δx = [ 2 - (-2) ] / n = 4/n           x_i = -2 + i Δx = -2 + 4 i /n

        l'aire =  Sigma f(x_i) *     Δx = 
    [tex]f(x)=4-x^2,\ \ \ x \in [-2, \ 2]\\\\\Delta x\ soit\ \frac{2-(-2)}{n} =\frac{4}{n} ,\\\\ x_i=-2+i\Delta x=-2+\frac{4i}{n}\\\\L'aire= \lim_{n \to \infty} \Sigma_0^n\ f(x_i)*\Delta x= \lim_{n \to \infty} \Sigma_0^n [4-(-2 + \frac{4i}{n})^2]*\frac{4}{n} \\\\= \lim_{n \to \infty}\frac{4}{n} \Sigma_0^n [4-(4-\frac{8i}{n}+\frac{16i^2}{n^2})]\\\\ \lim_{n \to \infty} \frac{4}{n} [\frac{8}{n} \Sigma_0^n\ i -\frac{16}{n^2} \Sigma_0^n\ i^2][/tex]



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