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1-109.

WHICH IS BETTER? Part One

Below are different sets of rectangles to approximate the area under a curve for the same interval. Look at the three different sets of rectangles and decide which will best approximate the area under the curve of this function for .

  1. Explain why your choice will determine the best approximation for the area.

    Which graph is the closest to the exact area under the curve?

    Midpoint Rectangles

  2. Will left endpoint rectangles always be an underestimate for any function? Explain.

    Notice that the graphs in the diagrams above show an increasing function. That is, as increases, the -values increase. What if we graphed a decreasing function instead? Would left-endpoint rectangles be underestimates then?

First quadrant graph, increasing curve opening up, starting about 1 fourth up on y axis, & 4 equal width vertical shaded bars, bottom edges on the axis, left edge of first bar labeled, a, right edge of last bar labeled, b, with top left vertex of each bar, on the curve.

Left Endpoint Rectangles

First quadrant graph, increasing curve opening up, starting about 1 fourth up on y axis, & 4 equal width, vertical shaded bars, bottom edges on x axis, left edge of first bar labeled, a, right edge of last bar labeled, b, with midpoint of top edge of each bar, on the curve.

Midpoint Rectangles

First quadrant graph, increasing curve opening up, starting about 1 fourth up on y axis, with 4 equal width, vertical shaded bars, bottom edges on x axis, left edge of first bar labeled, a, right edge of last bar labeled, b, with top right vertex of each bar, on the curve.

Right Endpoint Rectangles