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5.2 – The Definite Integral5.2 – The Definite Integral

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Review

Evaluate1

1lim 1

n

ni

i

n n

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The Definite Integral

• If f is a continuous function defined for a ≤ x ≤ b.

|

a

|

b

| | |●

x1*

●

x2*

●

x3*

●

x4*

•Divide the interval [a, b] into n subintervals of equal width of Δx = (b – a) / n.

• Let x1*, x2*, …., xn* be random sample points from these subintervals so that xi* lies anywhere in the subinterval [xi-1, xi].

x

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The Definite Integral

|

a

|

b

| | |●

x1*

●

x2*

●

x3*

●

x4*

x

Then the definite integral of f from a to b is

*

1

( ) limnb

ia ni

f x dx f x x

1f x

2f x

…

Riemann Sum[Bernhard Riemann

(1826 – 1866)]The limit of a Riemann Sum as

n → ∞ from x = a to x = b.

Note: xi is usually taken on the left side, right side, or midpoint of each rectangle since it is impossible to find a pattern for random points.

The limit of a Riemann Sum as

n → ∞ from x = a to x = b.

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Left, Right, Midpoint Rules Left:

Right:

Midpoint:

1ix a i x

ix a i x

2 1

2i

ix a x

1,2,3,....i

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Examples1. Use the Midpoint Rule with the given value of n to approximate the integral. Round your answer to four decimal places.

0

sec / 3 , 6x n

2. Write the limit of a Riemann Sum as a definite integral on the given interval.

1

lim , 1, 51

ixn

ni i

ex

x

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Examples3. Use the form of the definition of the definite integral to evaluate the integral.

4 2

12 5x x dx

4. Express the integral as a limit of a Riemann Sum, but do not evaluate.

10

14lnx x dx

5. Evaluate the integrals by interpreting in terms of area.

2 2

24a x dx

3

13 2b x dx

10

05c x dx

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Basic Properties of the IntegralLet a, b, and c be constants and f and g be continuous functions on [a, b].

1. ( ) ( )b a

a bf x dx f x dx

2. ( ) 0a

af x dx

3. Definite integrals can be positive or negative.

4. Not all definite integrals can be interpreted in terms of area, but definite integrals can be used to determine area.

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Properties of the IntegralLet a, b, and c be constants and f and g be continuous functions on [a, b].

1.b

ac dx c b a

2.b b b

a a af x g x dx f x dx g x dx

3.b b

a ac f x dx c f x dx

4. ;c b c

a a bf x dx f x dx f x dx a b c

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Properties of the IntegralLet a, b, and c be constants and f and g be continuous functions on [a, b].

0b

af x dx 5. If f (x) ≥ 0 for a ≤ x ≤ b, then

b b

a af x dx g x dx 6. If f (x) ≥ g (x) for a ≤ x ≤ b, then

b

am b a f x dx M b a

7. If m ≤ f (x) ≤ M for a ≤ x ≤ b, then

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Examples6. Use the properties of integrals to verify the inequality without

evaluating the integrals.

2 2

1 15 1a x dx x dx

/ 2

/ 6sin

6 3b x dx

2

/ 2

0sin

8d x x dx

2 3

0Estimate 3 3 using property 7.c x x dx