Calculus

Use two rectangles of equal width to estimate the area between the graph of f(x) = x - cos(πx) and the x-axis on the interval [1, 5]. Evaluate the function at the mid-point of each rectangle to find each height.

a) 8
b) 12
c) 16
d) 20

thank you in advance.

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  1. use two rectangles on [1,5] means they both have width 2.
    So, the 2 subintervals are [1,3] and [3,5]
    The midpoints are thus at x=2,4
    So, take f(2) and f(4) for the heights.
    Your approximation is just the sum of those two rectangles.

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  2. Thanks but can you explain me more because I am so so so confuse

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  3. what? You cannot evaluate f(x) at x=2 and x=4?
    The height of the rectangles is
    f(2) = 2 - cos(2π) = 2-1 = 1
    f(4) = 4 - cos(4π) = 4-1 = 3
    Since the width of each rectangle is 2, the area
    a = 2*1 + 2*3 = 8

    Now study your text's examples again, and it should make more sense.
    If not, consider asking your teacher (or a tutor) for some one-on-one help.
    youtube also has videos explaining quadrature.

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