CABChp5ishBNTs5 - CalcAB Chp 5ishB Riemann Sums...

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CalcAB Chp 5ishB Riemann Sums Approximation of area under a curve NOTES 1. Given a function over a closed interval [a, b] Find the area under the curve using rectangles. In the example below, look at BOTH pictures because rectangles can be drawn where they are too short or too tall. Left-Endpoint Right-Endpoint (determines height (determines height of the rectangles) of the rectangles) Here, first rectangle starts at x= a Here, last rectangle is at x = b For this curve the rectangles are too short Now, the rectangles are too tall Endpoints are on the curve f(x) Endpoints are on the curve f(x) The overall area is too small LOWER SUM Overall area is too large UPPER SUM Riemann Sums uses rectangles to find area under the curve (approximation and exact). Need the area of each rectangle and then the sum of the areas. NOTE: Which endpoint (right or left) is used does not determine whether the area will be too small or too big; that depends on the concavity of the curve and the interval used. For example, concave down on the upside of a mountain shape means the left-endpoint makes lower sum approximation of the area. But, for a section of graph that is concave up on the down side of a valley means the left-endpoint makes the upper sum approximation of the area. NOTE: The more rectangles, the more accurate the area. With infinitely many rectangles LOWER SUM=UPPER SUM=EXACT 2.
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