Unit 5Integration

Section 1Antiderivatives and Indefinite Integration

Antiderivative: a function F for which __________________ for all x in I.

For example, to find a function F whose derivative is f(x)=3x2, you could conclude that F(x)=x3 because F'(x)=3x2

Note: F is AN antiderivative, not THE antiderivative.

Theorem: Representation of AntiderivativesIf F is an antiderivative of f on an interval I, then G is an

antiderivative of f on the interval I if and only if G is of the form

_________________________, for all x in I where C is a constant.

Constant of Integration: ____________________ in the family

of derivatives G(x)=F(x)+C

General Antiderivative: the family of functions represented by G

where _____________________

General Solution: ____________________

Differential Equation (in x and y): an equation that involves

__________________________________

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Example 1.1Find the general solution of the differential equation y'=2.

Solving Differential EquationsWrite in differential form: dy=f(x)dx

Antidifferentiation (Indefinite Integration): the operation of finding all solutions of a differential equation denoted by ∫

∫f(x)dx:

dx:

Basic Integration Rules

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Example 1.2Find the general solution of ∫3xdx

Example 1.3Rewrite before integrating:A. ∫1/x3(dx)

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B. ∫√x(dx)

C. ∫2sin(x)(dx)

Example 1.4A. ∫dx

B. ∫(x+2)dx

C. ∫(3x4-5x2+x)dx

Example 1.5

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Example 1.6

Example 1.7A. ∫2/√x(dx)

B. ∫(t2+1)2dt

C. ∫(x3+3)/x(dx)

D. ∫∛x(x-4)(dx)

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Particular Solution: an antiderivative in which you know the

_________________________

Initial Condition: the value of _____________ for one value of x

Example 1.8Find the general solution of F(x)=1/x2, x>0.Then find the particular solution that satisfies the initial condition F(1)=0

Example 1.9A ball is thrown upward with an initial velocity of 64 ft/sec from an initial height of 80 ft.

a. Find the position function giving the height s as a function of the time t.

b. b. When does the ball hit the ground?

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Section 2Area

Sigma Notation: the sum of n terms a1, a2, a3, ..., an is written as

where i is the index of summation, ai is the ith term of the sum, and the upper and lower bounds of summation are n and 1.

Example 2.1

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Properties of Summation

Summation Formulas

Example 2.2

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AreaA=bh is the definition of the area of a rectangleExhaustion Method: to find the area of conics (curved figures). Squeeze the area between an inscribed polygon and a circumscribed one

Example 2.3Use 5 rectangles to find 2 approximations of the area lying between the graph of f(x)=-x2+5 and the x-axis between x=0 and x=2.

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Upper and Lower SumsGiven a plane region bounded above by the graph of a nonnegative continuous function y=f(x), bounded below by the x-axis, and bounded by the vertical lines x=a and x=b, to approximate the area, subdivide the interval [a,b] into n subintervals of width ∆x=(b-a)/n. The Extreme Value Theorem guarantees the existence of a maximum and minimum in each subinterval.

f(mi)=minimum value of f(x) in the ith subintervalf(Mi)=maximum value of f(x) in the ith subinterval

Inscribed Rectangle: lies inside the subregionCircumscribed Rectangle: extends outside the subregion

Lower Sum: the sum of the areas of the inscribed rectangles

Upper Sum: the sum of the areas of the circumscribed rectangles

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Example 2.4Find the upper and lower sums for the region bounded by the graph of f(x)=x2 and the x-axis between x=0 and x=2.

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Example 2.5Find the area of the region bounded by the graph of f(x)=x3, the x-axis, and the vertical lines x=0 and x=1.

Example 2.6 Find the area of the region bounded by the graph of f(x)=4-x2, the x-axis, and the vertical lines x=1 and x=2.

Example 2.7Find the area of the region bounded by the graph of f(y)=y2 and the y-axis for 0≤y≤1.

Midpoint RuleUse the midpoint of the interval to approximate the area.

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Example 2.8Use the Midpoint Rule with n=4 to approximate the area of the region bounded by the graph of f(x)=sinx and the x-axis for 0≤x≤π.

Section 3Riemann Sums and Definite Integrals

Example 3.1Consider the region bounded by the graph of f(x)=√x and the x-axis for 0≤x≤1. Evaluate the limit

where ci is the right endpoint of the partition given by ci=i2/n2 and ∆xi is the width of the ith interval.

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Norm (of a partition): the width of the largest subinterval of a partition ∆, denoted by ||∆||Regular (partition): every subinterval is of equal width

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Theorem: Continuity Implies IntegrabilityIf a function f is continuous on the closed interval [a,b], then f is

________________ on [a,b]. That is, _____________ exists.

Example 3.2Evaluate the definite integral

∫−2

1

2 xdx

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Theorem: The Definite Integral as the Area of a RegionIf f is continuous and nonnegative on the closed interval [a,b], then the area of the region bounded by the graph of f, the x-axis, and the vertical lines x=a and x=b is

Example 3.3Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula.

a. ∫1

3

4dx

b. ∫0

3

( x+2 )dx

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c. ∫−2

2

√4−x2dx

Definitions of Two Special Definite Integrals

1. If f is defined at x=a, then

2. If f is integrable on [a,b], then

Example 3.4Evaluate:

a. ∫π

π

sin ( x )dx

b. ∫3

0

( x+2 )dx

Theorem: Additive Interval PropertyIf f is integrable on the three closed intervals determined by a, b, and c, thenExample 3.5 ∫−1

1

¿ x∨dx

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Theorem: Properties of Definite IntegralsIf f and g are integrable on [a,b] and k is a constant, then the functions kf and f ± g are integrable on [a,b], and1.

2.

Example 3.6Evaluate ∫

1

3

(−x2+4 x−3 )dx using each of the following values.

∫1

3

x2dx=263

∫1

3

xdx=4

∫1

3

dx=2

Theorem: Preservation of Inequality1. If f is integrable and nonnegative on the closed interval [a,b], then

2. If f and g are integrable on the closed interval [a,b] and f(x)≤g(x) for every x in [a,b], then

Section 4The Fundamental Theorem of Calculus

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Theorem: The Fundamental Theorem of CalculusIf a function f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then

Example 4.1a. ∫

1

2

(x2−3 )dx

b. ∫1

4

3√ x dx

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c. ∫0

π /4

sec2 xdx

Example 4.2

∫0

2

¿2 x−1∨dx

Example 4.3Find 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.

Theorem: Mean Value Theorem for IntegralsIf f is continuous on the closed interval [a,b], then there exists a number c in the closed interval [a,b] such that

Average Value of a Function on an Interval

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If f is integrable on the closed interval [a,b], then the average value of f on the interval is

Example 4.4Find the average value of f(x)=3x2-2x on the interval [1,4]

Example 4.5At different altitudes in Earth's atmosphere, sound travels at different speeds. The speed of sound s(x) (in meters per second) can be modeled by

-4x+341, 0≤x<11.5295, 11.5≤x<22

s(x)= 3/4x+278.5, 22≤x<323/2x+254.5, 32≤x<50-3/2x+404.5, 50≤x≤80

where x is the altitude in kilometers. What is the average speed of sound over the interval [0,80]?

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The Second Fundamental Theorem of Calculus

Example 4.6Evaluate the function F ( x )=∫

0

x

cos ( t )dt

at x=0, π/6, π/4, π/3, and π/2

Theorem: The Second Fundamental Theorem of CalculusIf f is continuous on an open interval I containing a, then, for every x in the interval,

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Example 4.7Evaluate ddx [∫0

x

√ t2+1dt ]

Example 4.8

Find the derivative of F ( x )=∫π /2

x3

cos (t )dt

Theorem: The Net Change TheoremThe definite integral of the rate of change of quantity F’(x) gives the total change, or net change, in that quantity on the interval [a,b]

Example 4.9A chemical flows into a storage tank at a rate of (180+3t) liters per minute, where t is the time in minutes and 0≤t≤60. Find the amount of the chemical that flows into the tank during the first 20 minutes.

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Displacement: net change in position

Total Distance Traveled: since velocity can be negative

Example 4.10The velocity (in feet per second) of a particle moving along a line is v(t)=t3-10t2+29t-20 where t is the time in seconds.a. What is the displacement of the particle on the time interval 1≤t≤5?

b. What is the total distance traveled by the particle on the time interval 1≤t≤5?

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Section 5Integration by Substitution

Theorem: Antidifferentiation of a Composite FunctionLet g be a function whose range is an interval I, and let f be a function that is continuous on I. If g is differentiable on its domain and F is an antiderivative of f on I, then

Letting u=g(x) gives du=g’(x)dx and

Example 5.1Find ∫(x2+1)2(2x)dx

Example 5.2Find ∫5cos(5x)dx

Constant Multiple Rule

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Example 5.3Find the indefinite integral∫x(x2+1)2dx

Change of Variables

Example 5.4∫√2x-1dx

Example 5.5∫x√2x-1dx

Example 5.6∫sin2(3x)cos(3x)dx

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Theorem: The General Power Rule for IntegrationIf g is a differentiable function of x, then

Equivalently, if u=g(x), then

Example 5.7a. ∫3(3x-1)4dx

b. ∫(2x+1)(x2+x)dx

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c. ∫3x2√x3-2dx

d. ∫(-4x)/(1-2x2)2dx

e. ∫cos2xsinxdx

Theorem: Change of Variables for Definite IntegralsIf the function u=g(x) has a continuous derivative on the closed interval [a,b] and f is continuous on the range of g, then

Example 5.8 ∫

0

1

x (x2+1)3dx

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Example 5.9 ∫

1

5 x√2x−1

dx

Theorem: Integration of Even and Odd FunctionsLet f integrable on the closed interval [a,-a]. 1. If f is an even function, then

2. If f is an odd function, then

Example 5.10

∫−π

2

π2

( sin3 xcosx+sinxcosx )dx

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Section 6Numerical Integration

Theorem: The Trapezoidal RuleLet f be continuous on [a,b]. The Trapezoidal Rule for

approximating ∫a

b

f ( x )dx is

Moreover, as n→∞, the right-hand side approaches ∫a

b

f ( x )dx.

Example 6.1Use the Trapezoidal Rule to approximate ∫

0

π

sin ( x )dx

Compare the results when n=4 and n=8.

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Theorem: Integral of p ( x )=A x2+Bx+CIf p ( x )=A x2+Bx+C, then

Theorem: Simpson’s RuleLet f be continuous on [a,b] and let n be an even integer.

Simpson’s Rule for approximating ∫a

b

f ( x )dx is

Moreover, as n→∞, the right-hand side approaches ∫a

b

f ( x )dx.

Example 6.2Use Simpson's Rule to approximate ∫

0

π

sin ( x ) dx

Compare the results for n=4 and n=8

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