math calc2

Riemann Sums

1. Let A be the area of the region bounded by the graph of f(x) = x2 over the interval [1, 3].

a) Use the Right-Hand Rule to set up and simplify the formula for the sum area Rn of n right-hand rectangles:

Rn =

n∑

i=1

f(a + i∆x)∆x, where ∆x = b − a

n

b) Use your formula for Rn to approximate A by n = 5, 10, 100, 1000, and 50,000 rectangles. Present your data in table form (as shown), rounded to three decimal places of accuracy.

n Rn

5

10

100

1,000

50,000

c) Find the precise area A by computing the limit A = lim n→∞

Rn.

2. Let A be the area of the region bounded by the graph of f(x) = 2x + 1 over the interval [0, 3].

a) Sketch the graph of f and shade the region at issue.

b) Compute A by making an elementary geometric argument. No calculus.

c) Compute A using the Right-Hand Rule.

� 3, 4 Let A be the area of the region bounded by the graph of f over the given interval.

a) Sketch the graph of f and shade the region at issue.

b) Compute A using the Right-Hand Rule.

3. f(x) = x2 − 6x + 10, [2, 5]

4. f(x) = 4 − x2, [0, 2]

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Left-Hand & Midpoint Rules

� 5–6 Let A be the area of the region bounded by the graph of f(x) = x2 + 1 over [−2, 2].

5. a) Sketch the graph of f together with 4 left-hand rectangles.

b) Use your picture to calculate L4.

6. a) Sketch the graph of f together with 4 midpoint rectangles.

b) Use your picture to calculate M4.

Solutions to Selected Problems

1. a) Rn = 26n2 + 24n + 4

3n2

b) Table

c) A = 26 3

2. a) Graph

b) A = 12

c) A = 12

3. A = 6

4. A = 16 3

5. a) Graph

b) L4 = 10

6. a) Graph

b) M4 = 9

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