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Riemann Sums Jim Wang Mr. Brose Period 6

Riemann Sums Jim Wang Mr. Brose Period 6. Approximate the Area under y = x² on [ 0,4 ] a)4 rectangles whose height is given using the left endpoint b)4

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Page 1: Riemann Sums Jim Wang Mr. Brose Period 6. Approximate the Area under y = x² on [ 0,4 ] a)4 rectangles whose height is given using the left endpoint b)4

Riemann Sums

Jim Wang

Mr. Brose

Period 6

Page 2: Riemann Sums Jim Wang Mr. Brose Period 6. Approximate the Area under y = x² on [ 0,4 ] a)4 rectangles whose height is given using the left endpoint b)4

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

Page 3: Riemann Sums Jim Wang Mr. Brose Period 6. Approximate the Area under y = x² on [ 0,4 ] a)4 rectangles whose height is given using the left endpoint b)4

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

Page 4: Riemann Sums Jim Wang Mr. Brose Period 6. Approximate the Area under y = x² on [ 0,4 ] a)4 rectangles whose height is given using the left endpoint b)4

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

Page 5: Riemann Sums Jim Wang Mr. Brose Period 6. Approximate the Area under y = x² on [ 0,4 ] a)4 rectangles whose height is given using the left endpoint b)4

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

Page 6: Riemann Sums Jim Wang Mr. Brose Period 6. Approximate the Area under y = x² on [ 0,4 ] a)4 rectangles whose height is given using the left endpoint b)4

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

Page 7: Riemann Sums Jim Wang Mr. Brose Period 6. Approximate the Area under y = x² on [ 0,4 ] a)4 rectangles whose height is given using the left endpoint b)4

e) Evaluate the integral directly

Insert in Calculator:

f(x) = 04 x² dx

Ans. = 64/3