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Just diving into integral calculus

  1. Jul 5, 2011 #1
    And the first thing I'm tackling is understanding the area of a shape by inscribed/circumscribed polygons.

    For example, the shape produced by the line: y = x + 1, the y axis, and the vertical line x = 2.

    Now, my understanding is that the area is the limit as n approaches infinity of the sum of the area of n inscribed rectangles.

    Can someone explain how this translates to actual math? How would I find the area of the above shape?
  2. jcsd
  3. Jul 5, 2011 #2


    Staff: Mentor

    Is the other boundary of your region the x axis? If that's the region, its shape is a trapezoid whose area is 4.

    Using the limit of a Riemann sum to get the value, you would divide the interval [0, 2] into some number of subintervals, and the use rectangles or some other shape to approximate the area above each subinterval.

    For example, if the interval is divided into 4 subintervals, and inscribed rectangles are used, the area estimate is found to be .5*1 + .5*1.5 + .5 * 2.0 + .5 * 2.5 = .5*(7) = 3.5.

    To get a better approximation, divide the interval into more subintervals.
  4. Jul 6, 2011 #3
    Is it a valid method for me to do F(2) - F(0) to arrive at that answer? Would I be applying the theorem correctly?

    [itex]F(x) = \frac{x^2}{2} + x[/itex]

    F(2) = 4
    F(0) = 0

    4-0 = 4

    Since we are bounded by x = 0 and x = 2?
  5. Jul 6, 2011 #4
  6. Jul 6, 2011 #5
    That's really powerful!
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