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Riemann Sum
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Formula
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Step 1 Step 2 Step 3 Riemann Sum
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We start by subdividing the interval [a,b] into n subintervals The width of the interval [a,b] is b-a the width of each subinterval is The subintervals are Riemann Sum
![We start by subdividing the interval [a,b] into n subintervals The width of the interval [a,b] is b-a the width of each subinterval is The subintervals are Riemann Sum](http://images.slideplayer.com./33/10104449/slides/slide_4.jpg "We start by subdividing the interval [a,b] into n subintervals The width of the interval [a,b] is b-a the width of each subinterval is The subintervals are Riemann Sum")
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Term-103
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The Definite Integral Sec 5.3
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Remark: Definition: the definite integral of ƒ over [a, b] The Definite Integral Example: Find the definite integral of ƒ(x) = x + 2 over [ -1, 1 ] Definition: the definite integral of ƒ over [a, b]
![Remark: Definition: the definite integral of ƒ over [a, b] The Definite Integral Example: Find the definite integral of ƒ(x) = x + 2 over [ -1, 1 ] Definition: the definite integral of ƒ over [a, b]](http://images.slideplayer.com./33/10104449/slides/slide_7.jpg "Remark: Definition: the definite integral of ƒ over [a, b] The Definite Integral Example: Find the definite integral of ƒ(x) = x + 2 over [ -1, 1 ] Definition: the definite integral of ƒ over [a, b]")
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Partition is called a partition of [a, b]. Example Is a partition of [0, 10]. Is a partition of [0, 9]. Is a partition of [0, 10]. Note that the length of subintervals are not the same
![Partition is called a partition of [a, b]. Example Is a partition of [0, 10].](http://images.slideplayer.com./33/10104449/slides/slide_8.jpg "Is a partition of [0, 9]. Is a partition of [0, 10]. Note that the length of subintervals are not the same.")
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Example Is a partition of [0, 10]. the largest of all the subinterval widths subinterval widths Partition Def:Norm of the partition
![Example Is a partition of [0, 10].](http://images.slideplayer.com./33/10104449/slides/slide_9.jpg "the largest of all the subinterval widths subinterval widths Partition Def:Norm of the partition.")
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Riemann sum for ƒ on the interval [a, b]. Riemann Sum Example: Find the Riemann sum for ƒ(x) = x + 2 over [ 0, 5 ]
![Riemann sum for ƒ on the interval [a, b].](http://images.slideplayer.com./33/10104449/slides/slide_10.jpg "Riemann Sum Example: Find the Riemann sum for ƒ(x) = x + 2 over [ 0, 5 ].")
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Definition: The Definite Integral the definite integral of ƒ over [a, b] Definition: the definite integral of ƒ over [a, b] Definition: the definite integral of ƒ over [a, b]
![Definition: The Definite Integral the definite integral of ƒ over [a, b] Definition: the definite integral of ƒ over [a, b] Definition: the definite integral of ƒ over [a, b]](http://images.slideplayer.com./33/10104449/slides/slide_11.jpg "Definition: The Definite Integral the definite integral of ƒ over [a, b] Definition: the definite integral of ƒ over [a, b] Definition: the definite integral of ƒ over [a, b]")
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Notation: the definite integral of ƒ over [a, b] Remark: The Definite Integral Remark:
![Notation: the definite integral of ƒ over [a, b] Remark: The Definite Integral Remark:](http://images.slideplayer.com./33/10104449/slides/slide_12.jpg "Notation: the definite integral of ƒ over [a, b] Remark: The Definite Integral Remark:")
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The Definite Integral
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Example: Evaluate the following integrals by interpreting each in terms of areas. The Definite Integral
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Example: Evaluate the following integrals by interpreting each in terms of areas. The Definite Integral
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Area under the curve the definite integral of f from a to b If you are asked to find one of them choose the easiest one. The Definite Integral
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Riemann sum for ƒ on the interval [a, b].
![Riemann sum for ƒ on the interval [a, b].](http://images.slideplayer.com./33/10104449/slides/slide_17.jpg "Riemann sum for ƒ on the interval [a, b].")
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Example: Evaluate the following integrals by interpreting each in terms of areas. The Definite Integral
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THE DEFINITE INTEGRAL Term-103
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Property (1) THE DEFINITE INTEGRAL Example:
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THE DEFINITE INTEGRAL Property (2)
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THE DEFINITE INTEGRAL Property (3)
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THE DEFINITE INTEGRAL Term-091
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THE DEFINITE INTEGRAL
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DEFINITION Example: Find the average value of the function over the interval [-2,2] Average Value
![DEFINITION Example: Find the average value of the function over the interval [-2,2] Average Value](http://images.slideplayer.com./33/10104449/slides/slide_28.jpg "DEFINITION Example: Find the average value of the function over the interval [-2,2] Average Value")
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Term-082
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