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Riemann Sums
Jim Wang
Mr. Brose
Period 6
Approximate the Area undery = x² on [ 0,4 ]
a) 4 rectangles whose height is given using the left endpoint
b) 4 rectangles whose height is given using the right endpoint
c) 4 rectangles whose height is given using the midpoint
d) 4 trapezoids
e) 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
1 2 3 4
Ans. = 84/4 or 21
d) 4 trapezoids
Area of Trapezoid:
½ h ( b1 + b2 )
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) = 04 x² dx
Ans. = 64/3