THS Step By Step Calculus Chapter 4

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Name: ___________________________________

Class Period: _____________________________

Throughout this packet – there will be blanks you are expected to fill in prior to coming to class. This packet follows your

Larson Textbook. Do NOT throw away! Keep in 3 ring-binder until the end of the course.

Chapter 4.1 Antiderivatives and Indefinite Integration

Definition of an Antiderivative: Representations of Antiderivatives: Differential Equations: Notations for Antiderivatives:

Steps for finding solution with Initial Conditions:

Find the general solution of ( )

and find particular solution that satisfies F(1)=0

Step 1: Set up Integral ( ) ∫

Step 2: Find general solution ( )

for x > 0

Step 3: Apply Initial Condition to find C ( )

,

C = 1

Step 4: Write particular solution ( )

THS Step By Step Calculus Chapter 4

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THS Step By Step Calculus Chapter 4

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4.2 Area

Sigma Notation

Summation Formulas

General Steps for finding area:

Step 1: Identify Area of One Rectangle: ( ) Step 2: Add them up ∑

Finding Areas Summary

Right Endpoint Left Endpoint Midpoint

Step 1: Identify number of intervals n n n

Step 2: Identify endpoints [a, b] [a, b] [a, b]

Step 3: Determine width of rectangle:

Step 4: Determine x-value to evaluate ( ) (

)

Step 5: Determine height of rectangle: ( ) ( )

Step 6: Write area of rectangle: ( ) ( ) ( )

Step 7: Add them up ∑

∑

∑

Upper Sum: Lower Sum: Limits of the Lower and Upper Sums

THS Step By Step Calculus Chapter 4

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Definition of the Area of a Region in the Plane:

THS Step By Step Calculus Chapter 4

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4.3 Riemann Sums and the Definite Integral

Mathematician:_______________ Definition of a Riemann Sum

Regular Partition Definition of a Definite Integral

Theorem: Continuity implies Integrability Partition vs n:

∑ ( )

⇔

∑ ( )

THS Step By Step Calculus Chapter 4

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Steps for Evaluating a Definite Integral as a Limit

Step 1 Identify [a,b]

Step 2: Define

Step 3: For right endpoint, define Step 4: Write general term for ( )

Step 5: Write definition: ∫ ( ) ∑ ( )

Step 6: Rewrite using n: ∫ ( ) ∑ ( )

Step 7: Substitute ( ) Step 8: Evaluate limit

The Definite Integral as an Area:

Properties of Definite Integrals

THS Step By Step Calculus Chapter 4

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4.4 The Fundamental Theorem of Calculus

1st Fundamental Theorem of Calculus

Steps for Using 1

st FTC

Step 1: Write definite integral equal to Antiderivative evaluated at a and b: ∫ ( ) ( )

Step 2: Evaluate F(x) at a and b ( ) ( ) ( )

DO NOT INCLUDE +C

Mean Value Theorem for Integrals:

Average Value of a Function in Interval:

2nd Fundamental Theorem of Calculus

Steps for using 2

nd FTC

Step 1: Evaluate function at upper limit and multiply by derivative of upper limit Step 2: Evaluate function at lower limit and multiply by derivative of lower limit Step 3: Subtract

General Form:

[∫ ( )

( )

( )] ( ( )) ( ) ( ( )) ( )

Definite Integral as Number vs Function:

THS Step By Step Calculus Chapter 4

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4.5 Integration by Substitution Antidifferentiation of Composite Function:

U-Substitution/Change of Variables

∫ ( ( )) ( ) ∫ ( ) ( )

Generalized Power Rule:

U Substitution for Definite Integrals:

Integrating Even/Odd Functions

THS Step By Step Calculus Chapter 4

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Steps for using U-Substitution for Indefinite Integrals ∫ ( ) ( )

Step 1: Identify u=g(x) Let ( )

Step 2: Find du ( ) ( )

Step 3: Rewrite integral using u and du. ∫ ( ) ( ) ∫

∫

No x-terms should remain!

Step 4: Take the Antiderivative

Step 5: Substitute g(x)=u

( )

Step 6: Check by differentiating

( ) ( ) ( ) ( )

Steps for using U-Substitution for Definite Integrals

∫ ( )

Step 1: Identify u=g(x) Let

Step 2: Find du

Step 3: Change limits of Integration: ( ) ( )

Step 4: Rewrite integral using u and du and limits ∫( )

∫

No x-terms should remain!

Step 4: Take the integral

∫

Step 5: Evaluate

THS Step By Step Calculus Chapter 4

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4.6 Numerical Integration Trapezoid Rule

Steps for Trapezoid Rule

∫

with n=4

Step 1: Identify number of intervals, n n=4 Step 2: Identify endpoints a=0, b=π

Step 3: Determine base of trapezoid

Step 4: Find all

Step 5: Find all ( ) ( ) Step 6: Make a chart of ( )

Step 7: Calculate ( ) ( )

Step 8: Add all areas ∑

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