Riemann Sums and Definite Integration y = 6 y = x ex: Estimate the area under the curve y = x 2 + 2 from x = 0 to 3 using 3 subintervals and right endpoints,

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Presentation transcript:

Riemann Sums and Definite Integration y = 6 y = x ex: Estimate the area under the curve y = x from x = 0 to 3 using 3 subintervals and right endpoints, left endpoints and midpoints … Right Endpoints Left Endpoints Midpoints f(x) = x So 3 rectangles of width 1 starting at x=0…

Definition of area under a curve: The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of areas of approximating rectangles: (with right end points) (with left end points) (with sample points)

Sigma Notation Means ‘Sum’ Start at i=1 End at i=n Right Endpoints Left Endpoints Midpoints

ex: Estimate the area under the curve using midpoints and 4 subintervals from 0 to 2… 1 21

Distances Problem: Find the distance traveled by an object during a given time period if the velocity of the object is known at all times… D = (v)(t) for constant v If v isn’t constant… t (sec)V(ft/sec) t (sec)

…so subintervals are (0,4) (4,8) (8,12) …midpoint values would then be 2, 6 & 10