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Integration – UNIT 4 REVIEW
Antiderivatives
Substitution
Area
Definite Integrals
Applications
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Antiderivative
A function F is an antiderivative of f on an interval I if
for all x in I.
Ex.
is an antiderivative of
Since
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Theorem
Let G be an antiderivative of f. Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant.
Notice
are all antiderivatives of
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The process of finding all antiderivatives of a function is called integration.
The Notation:
read “the integral of f,”
means to find all the antiderivatives of f.
Indefinite integral
Integral sign
Integrand
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Basic Rules
Rule
Example
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Basic Rules
Rule
Ex. Find the indefinite integral
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Initial Value Problem
To find a function F that satisfies the differential equation
and one or more initial conditions.
Ex. Find a function f if it is known that:
Gives C = 3
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8. Integration by Substitution
Method of integration related to chain rule differentiation.
Ex. Consider the integral:
Sub to get
Integrate
Back Substitute
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9. Integration by Substitution
Steps:
1. Pick u = f (x), often the “inside function.”
2. Compute
3. Substitute to express the integral in terms of u.
4. Integrate the resulting integral.
5. Substitute to get the answer in terms of x.
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Ex. Evaluate
Pick u, compute du
Sub in
Integrate
Sub in
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Ex. Evaluate
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Definite Integral
Let f be defined on [a, b]. If
exists for all xn, xn, …,xn in the n subintervals each with width (b - a)/n, then this limit is called the definite integral of f from a to b and is denoted:
b is the upper limit, a is the lower limit.
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13. Integrability of a Function
Let f be continuous on [a, b]. Then f is integrable on [a, b]; that is,
exists.
Geometric Interpretation R1 R3 R2
a b
Area of R1 – Area of R2 + Area of R3
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14. Evaluating the Definite Integral
Ex. Evaluate
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15. Properties of the Definite Integral
(c is a constant)
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16. Substitution for Definite Integrals
Ex. Evaluate
Notice limits change
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