The Definite Integral

When we find the area under a curve by adding rectangles, the answer is called a Rieman sum.subintervalpartitionThe width of a rectangle is called a subinterval.The entire interval is called the partition.Subintervals do not all have to be the same size.

If we use subintervals of equal length, then the length of a subinterval is:The definite integral is then given by:

Leibnitz introduced a simpler notation for the definite integral:Note that the very small change in x becomes dx.

IntegrationSymbollower limit of integration upper limit of integration integrandvariable of integration(dummy variable)

We have the notation for integration, but we still need to learn how to evaluate the integral.

After 4 seconds, the object has gone 12 feet.In section 6.1, we considered an object moving at a constant rate of 3 ft/sec.Since rate . time = distance:If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.

If the velocity varies:Distance:(C=0 since s=0 at t=0)After 4 seconds:The distance is still equal to the area under the curve!Notice that the area is a trapezoid.

What if:We could split the area under the curve into a lot of thin trapezoids, and each trapezoid would behave like the large one in the previous example.It seems reasonable that the distance will equal the area under the curve.

We can use anti-derivatives to find the area under a curve!

Riemann SumsSigma notation enables us to express a large sum in compact form

Calculus Date: 2/18/2014 ID Check Objective: SWBAT apply properties of the definite integralDo Now: Set up two related rates problems from the HW Worksheet 6, 10HW Requests: pg 276 #23, 25, 26, Turn in #28 E.CIn class: Finish Sigma notation Continue Definite IntegralsHW:pg 286 #1,3,5,9, 13, 15, 17, 19, 21, Announcements:There is something in every one of you that waits and listens for the sound of the genuine in yourself. It is the only true guide you will ever have. And if you cannot hear it, you will all of your life spend your days on the ends of strings that somebody else pulls. Howard Thurman MaximizeAcademicPotentialTurn UP! MAP

When we find the area under a curve by adding rectangles, the answer is called a Rieman sum.subintervalpartitionThe width of a rectangle is called a subinterval.The entire interval is called the partition.Subintervals do not all have to be the same size.

subintervalpartitionThe width of a rectangle is called a subinterval.The entire interval is called the partition.

Lets divide partition into 8 subintervals.Pg 274 #9 Write this as a Riemann sum. 6 subintervals

If we use subintervals of equal length, then the length of a subinterval is:The definite integral is then given by:

Leibnitz introduced a simpler notation for the definite integral:Note that the very small change in x becomes dx.

Note as n gets larger and larger the definite integral approaches the actual value of the area.

IntegrationSymbollower limit of integration upper limit of integration integrandvariable of integration(dummy variable)

Calculus Date: 2/19/2014 ID Check Objective: SWBAT apply properties of the definite integralDo Now: Bell Ringer QuizHW Requests: pg 276 #25, 26, pg 286 1-15 odds In class: pg 276 #23, 28 Continue Definite IntegralsHW:pg 286 #17-35 odds Announcements:There is something in every one of you that waits and listens for the sound of the genuine in yourself. It is the only true guide you will ever have. And if you cannot hear it, you will all of your life spend your days on the ends of strings that somebody else pulls. Howard Thurman MaximizeAcademicPotentialTurn UP! MAP

Bell Ringer Quiz (10 minutes)

Riemann SumsLRAM, MRAM,and RRAM are examples of Riemann sums

Sn =

This sum, which depends on the partition P and the choice of the numbers ck,is a Riemann sum for f on the interval [a,b]

Definite Integral as a Limit of Riemann SumsLet f be a function defined on a closed interval [a,b]. For any partition P of [a,b], let the numbers ck be chosen arbitrarily in the subintervals [xk-1,xk].

If there exists a number I such that

no matter how P and the cks are chosen, then f is integrable on [a,b] and I is the definite integral of f over [a,b].

Definite Integral of a continuous function on [a,b]Let f be continuous on [a,b], and let [a,b] be partitioned into n subintervals of equal length x = (b-a)/n. Then the definite integral of f over [a,b] is given by

where each ck is chosen arbitrarily in the kth subinterval.

Definite integral

This is read as the integral from a to b of f of x dee x or sometimes as the integral from a to b of f of x with respect to x.

Using Definite integral notation

The function being integrated is f(x) = 3x2 2x + 5 over the interval [-1,3]

Definition: Area under a curveIf y = f(x) is nonnegative and integrable over a closed interval [a,b], then the area under the curve of y = f(x) from a to b is the integral of f from a to b,

We can use integrals to calculate areas and we can use areas to calculate integrals.

Nonpositive regionsIf the graph is nonpositive from a to b then

Area of any integrable function

= (area above the x-axis) (area below x-axis)

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Integral of a ConstantIf f(x) = c, where c is a constant, on the interval [a,b], then

Evaluating Integrals using areasWe can use integrals to calculate areas and we can use areas to calculate integrals.

Using areas, evaluate the integrals:

1)

2)

Evaluating Integrals using areasEvaluate using areas:

3)

4) (a

Evaluating integrals using areasEvaluate the discontinuous function:

Since the function is discontinuous at x = 0, we must divide the areas into two pieces and find the sum of the areas

= -1 + 2 = 1

Integrals on a CalculatorYou can evaluate integrals numerically using the calculator. The book denotes this by using NINT. The calculator function fnInt is what you will use.

= fnInt(xsinx,x,-1,2) is approx. 2.04

Evaluate Integrals on calculatorEvaluate the following integrals numerically:

= approx. 3.14

= approx. .89

Rules for Definite IntegralsOrder of Integration:

Rules for Definite IntegralsZero:

Rules for Definite IntegralsConstant Multiple:

Any number k

k= -1

Rules for Definite Integrals4) Sum and Difference:

Rules for Definite Integrals5) Additivity:

Rules for Definite IntegralsMax-Min Inequality: If max f and min f are the maximum and minimum values of f on [a,b] then:

min f (b a) max f (b a)

Rules for Definite IntegralsDomination: f(x) g(x) on [a,b]

f(x) 0 on [a,b] 0 (g =0)

Using the rules for integrationSuppose:

Find each of the following integrals, if possible: b) c)

d) e) f)

Calculus Date: 2/27/2014 ID Check Obj: SWBAT connect Differential and Integral CalculusDo Now:

http://www.youtube.com/watch?v=mmMieLl-Jzs HW Requests: 145 #2-34 evens and 33HW: Complete SM pg 156, pg 306 #1-19 odds Announcements:Mid Chapter Test Fri. Sect. 6.1-6.3Careful of units, meaning of area, asymptotes, properties of integrals

Handout InversesSaturday Tutoring 10-1 (limits)There is something in every one of you that waits and listens for the sound of the genuine in yourself. It is the only true guide you will ever have. And if you cannot hear it, you will all of your life spend your days on the ends of strings that somebody else pulls. Howard ThurmanMaximizeAcademicPotential

The Fundamental Theorem of Calculus, Part IAntiderivativeDerivative

Applications of The Fundamental Theorem of Calculus, Part I1.

2.

Applications of The Fundamental Theorem of Calculus, Part I

Applications of The Fundamental Theorem of Calculus, Part I

Applications of The Fundamental Theorem of Calculus, Part IFind dy/dx.

y =

Since this has an x on both ends of the integral, it must be separated.

Applications of The Fundamental Theorem of Calculus, Part I

=

Applications of The Fundamental Theorem of Calculus, Part I

=

=

The Fundamental Theorem of Calculus, Part 2If f is continuous at every point of [a,b], and if F is any antiderivative of f on [a,b], then

This part of the Fundamental Theorem is also called the Integral Evaluation Theorem.

Applications of The Fundamental Theorem of Calculus, Part 2

End here

Calculus Date: 2/27/2014 ID Check Obj: SWBAT connect Differential and Integral CalculusDo Now:

http://www.youtube.com/watch?v=mmMieLl-Jzs HW Requests: 145 #2-34 evens and 33HW: SM pg 156 Announcements:Mid Chapter Test Fri. Sect. 6.1-6.3Careful of units, meaning of area, asymptotes, properties of integrals

Handout InversesSaturday Tutoring 10-1 (limits)There is something in every one of you that waits and listens for the sound of the genuine in yourself. It is the only true guide you will ever have. And if you cannot hear it, you will all of your life spend your days on the ends of strings that somebody else pulls. Howard ThurmanMaximizeAcademicPotential

The Fundamental Theorem of Calculus, Part IAntiderivativeDerivative

The Fundamental Theorem of Calculus, Part 2If f is continuous at every point of [a,b], and if F is any antiderivative of f on [a,b], then

This part of the Fundamental Theorem is also called the Integral Evaluation Theorem.

AntidifferentiationA function F(x) is an antiderivative of a function f(x) if F(x) = f(x) for all x in the domain of f. The process of finding an antiderivative is called antidifferentiation. If F is any antiderivative of f then= F(x) + C If x = a, then 0 = F(a) + C C = -F(a)= F(x) F(a)

Using the rules for definite integralsShow that the value of is less than 3/2

The Max-Min Inequality rule says the max f . (b a) is an upper bound.The maximum value of (1+cosx) on [0,1] is 2 so the upper bound is: 2(1 0) = 2 , which is less than 3/2

Average (Mean) Value

Applying the Mean ValueAv(f) =

= 1/3(3) = 1

4 x2 = 1 when x = 3 but only 3 falls in the interval from [0,3], so x = 3 is the place where the function assumes the average.Use fnInt

Mean Value Theorem for Definite IntegralsIf f is continuous on [a,b], then at some point c in [a,b],

Trapezoidal RuleTo approximate , use

T = (y0 + 2y1 + 2y2 + . 2yn-1 + yn)

where [a,b] is partitioned into n subintervals of equal length h = (b-a)/n.

Using the trapezoidal ruleUse the trapezoidal rule with n = 4 to estimate

h = (2-1)/4 or , so

T = 1/8( 1+2(25/16)+2(36/16)+2(49/16)+4) = 75/32 or about 2.344

Simpson RuleTo approximate , use

S = (y0 + 4y1 + 2y2 + 4y3. 2yn-2 +4yn-1 + yn)

where [a,b] is partitioned into an even number n subintervals of equal length h =(b a)/n.

Using Simpsons RuleUse Simpsons rule with n = 4 to estimate

h = (2 1)/4 = , so

S = 1/12 (1 + 4(25/16) + 2(36/16) + 4(49/16) + 4) = 7/3

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