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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 ( )
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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
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THS Step By Step Calculus Chapter 4
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Definition of the Area of a Region in the Plane:
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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:
∑ ( )
⇔
∑ ( )
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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
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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:
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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
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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
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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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