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16.7 Area Under a Curve

16.7 Area Under a Curve. (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.

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16.7 Area Under a Curve

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  1. 16.7 Area Under a Curve

  2. (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.

  3. (Write now) (difficult task!) 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) a b 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

  4. Let’s learn by doing! Ex 1) Find an approximation of the area of the region bounded by f (x) = 2x2 , y = 0, x = 1, and x = 3 by: a) circumscribed rectangles with a width of 0.5. draw sketch 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 IV III II I 1 1.5 2.5 3 2 This is an overestimation! Why?

  5. Ex 1) Find an approximation of the area of the region bounded by f (x) = 2x2 , y = 0, x = 1, and x = 3 by: b) inscribed rectangles with a width of 0.5. draw sketch 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 1 1.5 2.5 3 2 This is an underestimation! Why? *Note: the actual area will be some value between 13.5 and 21.5

  6. 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

  7. Homework #1607 Pg 892 #7 – 11 all

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