1. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Integral
chapter6
The Indefinite Integral
Substitution
The Definite Integral As a Sum
The Definite Integral As Area
The Definite Integral: The Fundamental Theorem of Calculus
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Antiderivative
An antiderivative of a function f is a function F such that
Ex.
An antiderivative of is
since
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3. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Indefinite Integral
The expression:
read “the indefinite integral of f with respect to x,”
means to find the set of all antiderivatives of f.
x is called the variable of integration
Integrand
Integral sign
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Constant of Integration
Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant.
Notice
Represents every possible antiderivative of 6x.
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Power Rule for the Indefinite Integral, Part I
Ex.
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Power Rule for the Indefinite Integral, Part II
Indefinite Integral of ex and bx
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7. Sum and Difference Rules
Ex.
Constant Multiple Rule
Ex.
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8. Integral Example/Different Variable
Ex. Find the indefinite integral:
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Position, Velocity, and Acceleration Derivative Form
If s = s(t) is the position function of an object at time t, then
Velocity = v =
Acceleration = a =
Integral Form
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10. Integration by Substitution
Method of integration related to chain rule differentiation. If u is a function of x, then we can use the formula
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11. Integration by Substitution
Ex. Consider the integral:
Sub to get
Integrate
Back Substitute
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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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Ex. Evaluate
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Shortcuts: Integrals of Expressions Involving ax + b
Rule
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Riemann Sum
If f is a continuous function, then the left Riemann sum with n equal subdivisions for f over the interval [a, b] is defined to be
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17. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Definite Integral
If f is a continuous function, the definite integral of f from a to b is defined to be
The function f is called the integrand, the numbers a and b are called the limits of integration, and the variable x is called the variable of integration.
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Approximating the Definite Integral
Ex. Calculate the Riemann sum for the
integral using n = 10.
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The Definite Integral
is read “the integral, from a to b of f(x)dx.”
Also note that the variable x is a “dummy variable.”
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20. The Definite Integral As a Total
If r(x) is the rate of change of a quantity Q (in units of Q per unit of x), then the total or accumulated change of the quantity as x changes from a to b is given by
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21. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Definite Integral As a Total
Ex. If at time t minutes you are traveling at a rate of v(t) feet per minute, then the total distance traveled in feet from minute 2 to minute 10 is given by
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22. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Area Under a Graph
Width:
(n rect.)
a b
Idea: To find the exact area under the graph of a function.
Method: Use an infinite number of rectangles of equal width and compute their area with a limit.
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Approximating Area
Approximate the area under the graph of
using n = 4.
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Area Under a Graph
a b
f continuous, nonnegative on [a, b]. The area is
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25. Geometric Interpretation (All Functions)
a b R2
Area of R1 – Area of R2 + Area of R3
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26. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Area Using Geometry
Ex. Use geometry to compute the integral
Area =4
Area = 2
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27. Fundamental Theorem of Calculus
Let f be a continuous function on [a, b].
2. If F is any continuous antiderivative of f and is defined on [a, b], then
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28. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Fundamental Theorem of Calculus
Ex.
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29. Evaluating the Definite Integral
Ex. Calculate
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30. Substitution for Definite Integrals
Ex. Calculate
Notice limits change
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31. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Computing Area
Ex. Find the area enclosed by the x-axis, the vertical lines x = 0, x = 2 and the graph of
Gives the area since 2x3 is nonnegative on [0, 2].
Antiderivative
Fund. Thm. of Calculus
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