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1-58.

If for , estimate the area under the curve for as follows.  Homework Help ✎

  1. Using six left endpoint rectangles. The first two rectangles are drawn for you.

    Regarding the word ESTIMATE: You are not being asked to make a guess. You still need to compute. Of course, the area using rectangles is an approximation of the actual area under .

    Sketch the remaining rectangle. Notice: The height of the LAST rectangle is evaluated at (not ).

    Each rectangle has a base of and a height of .  ___________?

  2. Using six right endpoint rectangles.

    Notice: The height of the FIRST rectangle is evaluated at . The height of the LAST rectangle is evaluated at .

    Each rectangle has a base of and a height of .  __________?

  3. Using six trapezoids. What do you notice? Does this always happen?

    ?

    The trapezoidal sum is the AVERAGE of the left- and right-endpoint sums! Expain why this happens (algebraically and geometrically).

Left-end rectangles under a parabola

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