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16.7 Area Under a Curve

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(Don’t write ) We have been emphasizing the connection between the derivative and slopes. There is another fundamental concept in Calculus we will lay the foundation for today. The idea of area under a curve. It is easy to find the area under a straight line (could be a rectangle, or triangle, or trapezoid), but what if it is curved or has lots of curves? The underlying idea comes from estimating the area using skinny (and then even more skinny) rectangles.

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(Write now) a b Find the area under this curve. We can estimate it! Say we divided it into 4 parts & find areas of RECTANGLES A = b · h (*We can do as many as we want, but our practice problems usually have 4) (difficult task!) Using “inscribed rectangles” - rectangles are completely “under” the curve - for this graph, the rectangle heights are obtained from the left side of each strip (referred to as LRAM) OR Using “circumscribed rectangles” - rectangles are somewhat outside the curve - for this graph, the rectangle heights are obtained from the right side of each strip (referred to as RRAM) Note: In Calculus, these rectangle widths will get smaller & smaller. We would get a limit of the sum of all the areas. left rectangle approx method

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Area ≈ base height of 4 rectangles Area ≈ (0.5)(f (1.5)) + (0.5)(f (2)) + (0.5)(f (2.5)) + (0.5)(f (3)) OR ≈ (0.5)(f (1.5) + f (2) + f (2.5) + f (3)) ≈ (0.5)(4.5 + 8 + 12.5 + 18) ≈ (0.5)(43) ≈ 21.5 Let’s learn by doing! Ex 1) Find an approximation of the area of the region bounded by f (x) = 2x 2, y = 0, x = 1, and x = 3 by: a) circumscribed rectangles with a width of 0.5. draw sketch 1 3 2 1.52.5 This is an overestimation! Why? I II III IV

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Same process as before... Area ≈ (0.5)(f (1) + f (1.5) + f (2) + f (2.5)) ≈ (0.5)(2 + 4.5 + 8 + 12.5) ≈ (0.5)(27) ≈ 13.5 Ex 1) Find an approximation of the area of the region bounded by f (x) = 2x 2, y = 0, x = 1, and x = 3 by: b) inscribed rectangles with a width of 0.5. draw sketch 1 3 2 1.52.5 This is an underestimation! Why? *Note: the actual area will be some value between 13.5 and 21.5

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Note: When the graph was concave up, inscribed rectangles used the left sides & circumscribed rectangles used the right sides. What if it was concave down? Draw a sketch! *This is why the sketch is so important! Don’t memorize – SKETCH! inscribed uses right sides circumscribed uses left sides

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Homework #1607 Pg 892 #7 – 11 all

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