The Fundamental Theorem of Calculus. If f is continuous on the closed interval [a,b] and F is an...

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The Fundamental Theorem of Calculus

Transcript of The Fundamental Theorem of Calculus. If f is continuous on the closed interval [a,b] and F is an...

Page 1: The Fundamental Theorem of Calculus. If f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then.

The Fundamental Theorem of Calculus

Page 2: The Fundamental Theorem of Calculus. If f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then.

The Fundamental Theorem of Calculus

b

a

aFbFdxxf )()()(

If f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then

Page 3: The Fundamental Theorem of Calculus. If f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then.

Some Notes

Provided you can find the antiderivative of f you can now find the definite integral without limits. (YAY!)

Notation:

You don’t need the constant of integration:

)()()()( aFbFxFdxxfb

a

b

a

)()(

)()(

)()(

aFbF

CaFCbF

CxFdxxfb

a

b

a

Page 4: The Fundamental Theorem of Calculus. If f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then.

Examples

2

1

2 3 dxx

4

1

3 dxx

4

0

2sec

xdx

Page 5: The Fundamental Theorem of Calculus. If f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then.

Examples

2

0

12 dxx

4

0

2 dttt

2

0

22 cossin

dxxx

Page 6: The Fundamental Theorem of Calculus. If f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then.

Examples

2

2

cos2

dttt

Page 7: The Fundamental Theorem of Calculus. If f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then.

Examples

Find the area of the region bounded by the graph of y=2x2-3x+2, the x-axis, and the vertical lines x=0 and x=2

Page 8: The Fundamental Theorem of Calculus. If f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then.

The Mean Value Theorem for Integrals

Rectangle Curve der the Un Rectange

bedCircumscri of AreaRegion theof Area Inscribed of Area

We know

The mean value theorem states that somewhere “between” the two there is a rectangle whose area is precisely equal to the area of the region under the curve.

Page 9: The Fundamental Theorem of Calculus. If f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then.

The Mean Value Theorem for Integrals

))(()( abcfdxxfb

a

If f is continuous on [a,b], then there exists a number c on [a,b] such that

f (c) is the average value of f on [a,b]

b

a

dxxfab

)(1

valueAverage

Page 10: The Fundamental Theorem of Calculus. If f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then.

Example]4,1[on 23)( of valueaverage theFind 2 xxxf

Note: The area under the graph of f is equal to the area of the rectangle whose height is the average value of f


Let f I x y in the interval I f x

Let f I x y in the interval I f x

Antiderivatives and Initial Value Problems...Antiderivatives and Initial Value Problems Definition: An antiderivative of a function f on an interval [a,b] is another function F such

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Chapter 5 Integration. Indefinite Integral or Antiderivative.

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New MergedFile · 2019. 11. 12. · If fis continuous on the interval [a, b] and F(x) is an antiderivative off, that is F'(x) =f(x), then Example 1 Compute 2xdx numerically and using

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Lab 6-5 Weather Patterns Map A: Isotherms Connect points of equal value Connect points of equal value Use an interval of 10°F Use an interval of 10°F.

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32 Chorales & Tuning Exercises in F Interval Tuning 1 ...

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Class Interval Class Boundaries Class Mark, x i Frequency, f i Relative Freq’y. %

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5.1 Antidifferentiation Reversing differentiation Integration · Theorem 5.1 Antiderivatives of a function differ by a constant If F is an antiderivative of the continuous function

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MAT01A1: Antiderivatives · are also antiderivatives of f(x) = 2xbecause G0(x) = 2x and H0(x) = 2x. Theorem: If Fis an antiderivative of f on an interval I, then the most general

Miss Battaglia AP Calculus AB/BC. Definition of Antiderivative A function F is an antiderivative of f on an interval I if F’(x)=f(x) for all x in I. Representation.

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MANUAL-Interval, Interval 120cm, Interval 140cm-ALL-vA

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Chapter Four: Integration 4.1 Antiderivatives and ... · 4.1 Antiderivatives and Indefinite Integration Definition of Antiderivative – A function F is an antiderivative of f on

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INTEGRALS 5. Indefinite Integrals INTEGRALS The notation ∫ f(x) dx is traditionally used for an antiderivative of f and is called an indefinite integral.

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abdullaeid.net · Inde nite Integrals If F(x) is the antiderivative of f (x), we will write Z f (x)dx = F(x) |{z} antiderivative +C where The symbol R is called the integral sign.

abdullaeid.net · Inde nite Integrals If F(x) is the antiderivative of f (x), we will write Z f (x)dx = F(x) |{z} antiderivative +C where The symbol R is called the integral sign.

Solving Definite Integrals - Virginia · PDF fileSolving Definite Integrals Theorem: (Fundamental Theorem I) Or: If F is an antiderivative for f, then We determined using Simpson’s

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MAT01A1: Integration · Antiderivatives A function Fis an antiderivative of fon an interval Iif F0(x) = f(x) for all x2I. Example: F(x) = x2 is an antiderivative of f(x) = 2x. Question:

MAT01A1: Integration · Antiderivatives A function Fis an antiderivative of fon an interval Iif F0(x) = f(x) for all x2I. Example: F(x) = x2 is an antiderivative of f(x) = 2x. Question:

Let: Find the an antiderivative

Let: Find the an antiderivative