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