Calculus-Approximate areas

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Estimate the area under the graph of f(x)= x^2 + 3 x from x=1 to x=10 using the areas of 3 rectangles of equal width, with heights of the rectangles determined by the height of the curve at

a) left endpoints:
b) right endpoints:

  • Calculus-Approximate areas -

    so the values of x would be
    x = 1,4,7, and 10 , equal width = 3

    left end-points
    x = 1 , 4, 7
    f(1) = 1+3 = 4
    f(4) = 16+12 = 28
    f(7) = 49+21 = 70
    area = 3(4) + 3(28) + 3(70) = 306 ---> under-estimation

    right end-points
    x = 4 , 7, and 10
    f(4) = 28
    f(7) = 70
    f(10) = 130
    area = 3(28+70+130) = 684 ---> over-estimation

    let's average them:
    (306+684)/2 = 495

    which is close to what if we had taken the midpoint of the base of each rectangle
    f(2.5) = 13.75
    f(5.5) = 46.75
    f(8.5) = 97.75
    area = 3(13.75+46.75+97.75) = 474.75

    exact answer by Calculus:
    481.5

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