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The function is graphed at right, along with four left endpoint rectangles which approximate the area under the curve from to .

  1. Why does it look like there are only three rectangles?

    What happens if the left endpoint is on the -axis?

    Since , the height of the rectangle is .

  2. Recall that area under the -axis is negative, while area above the -axis is positive. Approximate the area under the curve for using these four rectangles.

    Sum the areas of all four rectangles together. Remember to treat a rectangle's area as negative if it is below the -axis.

Increasing cubic curve, y = x cubed + 1, & 3 shaded vertical rectangular bars of width 1, first bar, top left vertex at (negative 2, comma. 0), & bottom left vertex at (negative 2, comma negative 7), second bar, bottom left vertex at the origin, top left vertex at (0, comma 1), third bar, bottom let vertex at (1, comma 0), top left vertex at (1, comma 1).