Jim Wang Mr. Brose Period 6

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Presentation transcript:

Jim Wang Mr. Brose Period 6 Riemann Sums Jim Wang Mr. Brose Period 6

Approximate the Area under y = x² on [ 0,4 ] 4 rectangles whose height is given using the left endpoint 4 rectangles whose height is given using the right endpoint 4 rectangles whose height is given using the midpoint 4 trapezoids Evaluate the integral directly

a) 4 rectangles whose height is given using the left endpoint Left: ( b • h ) (1)0 + (1)1 + (1)4 + (1)9 1 2 3 4 Ans. = 14

b) 4 rectangles whose height is given using the right endpoint Right: ( b • h ) (1)1 + (1)4 + (1)9 + (1)16 1 2 3 4 Ans. = 30

c) 4 rectangles whose height is given using the midpoint Midpoint: ( b • h) (1)1/4 + (1)9/4 + (1)25/4 + (1)49/4 1 2 3 4 Ans. = 84/4 or 21

d) 4 trapezoids Area of Trapezoid: ½ h ( b1 + b2 ) 1 2 3 4 ½ [ 44 ] ½ (1) [ 0+1 + 1+4 + 4+9 + 9+16 ] 1 2 3 4 ½ [ 44 ] Ans. = 22

e) Evaluate the integral directly Insert in Calculator: f(x) = 04 x² dx Ans. = 64/3