Big idea Feedback & Assessment of Your Successabout the function itself? ex. you know velocity function but what to know the _____ function. A function F is called an antiderivative

1 Calculus AP U5 – Integration (AP) Name: ________________

1

Big idea Calculus is an entire branch of mathematics. Calculus is built on two major complementary ideas. The first is differential calculus, which is concerned with the instantaneous rate of change. This can be illustrated by the slope of a tangent to a function's graph. The second is integral calculus, which studies the areas under a curve. These two processes act inversely to each other. Calculus allows you to find optimal solutions to mathematical expressions and is used in medicine, engineering, economics, computer science, business, physical sciences, statistics, and many more areas.

Feedback & Assessment of Your Success

Finished

assignment

pages?

Made

corrections?

Summarized

notes in a

journal?

Added your

own

explanations?

How many

extra

practice

questions

did you try

in each

topic?

Tentative TEST date:

______________________

Date Pages Topics Questions to ask the

teacher:

1.5days

2-4 Antiderivatives & Indefinite Integration (AP) Journal #1

3days

5-11 Estimating Areas – Numerical Integration (AP) Journal #2

1.5days 12-14

Sigma Limits of Finite Sums (AP) Journal #3

15-17

Definite Integrals (AP) Journal #4

2days 18-21

FTC & Average Value (AP) Journal #5

3days

22-27 More FTC & MVT for Integrals (AP) Journal #6

3days 28-33

Interpret Integrals (AP) Journal #7


2

ASSIGNMENT Antiderivatives & Indefinite Integration (AP)

1. So far, given a function, we know how to find a

rate of change using the _____________________,

but what if all we knew was how a function was

changing with time, and we wanted to find out

about the function itself? ex. you know velocity

function but what to know the _______________

function.

A function F is called an antiderivative of f on

an interval I if )()( xfxF =′ for all x in I

Theorem: If F is an antiderivative of f on an

interval I , then the most general antiderivative of

f on I is

CxF +)(

where C is an arbitrary constant.

2. Graph a possible ( )F x for the given graph of

( )F x′

Explain why there are infinitely many answers:

3. Find the antiderivatives

a) 3

)( xxf =

b) 2

)(−= xxf

c) 132)(8 ++= xxxf

d) 2

1

2sin5)(−

+= xxxf

e) 6

4

1tansec7)(

−+= xxxxf

f) 2

1

38)(

xxxf

−−=

g) If 21

5cos8)(

xxxf

+−=′ , find )(xf

4. Different notation: Find the indefinite integrals

a) 4 dx

b)

23 x dx

c) ( 1)x x dx+

d) 4

34 x dx

x

+

e)

22 1

x x

dxx

+ +

f)

g)


3

In each of the graphs below, determine which curve is )(xf , and which curve is the antiderivative ( )F x .

5.

6.

7. A differential equation is an equation explicitly

solved for a derivative of a particular equation.

Solving a differential equation involves finding the

original function from which the derivative came.

The _______________ solution involves +C . The

_________________ solution uses an

____________________________________ to find

the specific value of C.

A separable differential equation is one where it is

possible to separate all the x and y variables.

• Take Leibniz Form: ( )dy

f xdx

′=

• Change to Differentiable Form: ( )dy f x dx′=

• Perform antiderivative or indefinite integration

operation: ( )dy f x dx′=

8. Sometimes, it is difficult or impossible to find the

antiderivative of a function, but we can still gather

info about it graphically.

A direction field, which shows the slope at given

points, can be used to sketch a graph of the

antiderivative of a function.

Ex. If )1(sin)( +⋅=′ xxxf and 1)0( −=f ,

sketch )(xf .

Solve the differential equations.

9. Given 4 3

dyx

dx= + and (1) 6y = , find the

equation for y

10.

If 3

53

)(x

xxxf

+=′ and

3

1)1( =f , find )(xf


4

11. If xxxf +=′′ sin)( with 7)0( =f and

2)0( =′f , find )(xf

12.

Find y(x).

13. Suppose the rate of change of concentration of a

vitamin in the bloodstream at time t is given by

te

dt

dc 3.01.0

−−=

If there is initially 1mg of the vitamin in the

bloodstream, then what is the concentration as a

function of time?

14. A ball is thrown upward with a speed of 10 m/s

from a building that is 30m tall. Find a formula

describing the height of the ball above the ground t

seconds later.


5

ASSIGNMENT Estimating Areas – Numerical Integration (AP)

1. A scout moves north in the forest for a distance of

50m in 150sec, stops for 30sec, then moves south

80m in 160sec.

a. Sketch a displacement-time graph

b. Sketch a velocity-time graph

c. Find the area under the velocity-time graph.

d. What does the slope of d-t graph represent?

e. What does the area under the v-t graph

represent?

f. What would the slope of v-t graph represent?

g. What would the area under the a-t graph

represent?

2.

a. Find the instantaneous rate of change at t=1.

What does it represent?

b. Find the average acceleration from t=2 to t=6.

c. Find the distance travelled from t=2 to t=6

d. Sketch d-t graph


6

Calculus answers two very important questions. The first, how to find the instantaneous rate of change, we answered

with our study of the derivative. We are now ready to answer the second question: how to find the area of irregular

regions.

3. Goal: To find the area of the shaded region R that

lies above the x-axis, below the graph of 2

1y x= − and between the vertical lines x = 0

and x = 1.

Left Riemann Approximation Method: LRAM

0 1 1

1

0

( ) ( ) ( )

( )

n n

n

k

k

L f x x f x x f x x

x f x

−

−

=

= ∆ + ∆ + + ∆

= ∆

�

we divide the interval ],[ ba into n subintervals of

equal width b a

xn

−∆ = and k

x a k x= + ∆

4. Right Riemann Approximation Method: RRAM

1 2

1

( ) ( ) ( )

( )

n n

n

k

k

R f x x f x x f x x

x f x=

= ∆ + ∆ + + ∆

= ∆

�

Mid Riemann Approximation Method: MRAM

1

1

0

( )2

n

k kn

k

x xM x f

−+

=

+= ∆


7

5. For continuous functions f ,

Definite integral of f from a to b is

1

( ) lim ( )n

b

ka n

k

f x dx f x x→∞

=

= ∆where the interval ],[ ba is divided

into n subintervals of equal width

then let n → ∞

Notes:

• x is a ______________ variable

( ) ( )b b

a af x dx f t dt=

• b

adxxf )( is a ____________

called ____________________

while

( )f x dx is a ___________

called ____________________

• b

adxxf )( represents

_______ accumulation over an

interval ],[ ba not ________

since area is always positive!

• If you do think of the integral

as area under curve, then keep

the following in mind:

6. Velocity function of a particle moving left/right on

a horizontal line is 3

2( ) cos 1.5V t t t= + on [ ]0,5 .

a) What does negative velocity on [ ]2,4.5

mean for position of a particle?

b) Find the Right Riemann sum using ten

subintervals.

c) The sum you found, what does it

represent?

d) The actual value of the “area under” is

-2.887m what is the % error in your

answer?


8

7. The velocity, m/s2, function of a projectile fired straight up into the air is ( ) 160 9.8f t t= − .

a) Use the Left Riemann sum with six subintervals to estimate how far the projectile rises during the

first 3 sec.

b) How close (% error) do the sums come to the actual value of 435.9 m?

8. Use the midpoint rule to find an approximation to

+

2

1

31

1dx

x using n = 4

9. Use the midpoint rule to approximate

−

+

1

2

25dxx using n = 3


9

Trapezoidal Rule

)]()(2)...(2)(2)([2

)( 1210 nn

b

a

n xfxfxfxfxfx

Tdxxf ++++∆

=≈ −

where n

abx

−=∆ and

kx a k x= + ∆

Simpson’s Rule

)]()(4)(2)...(4)(2)(4)([3

)( 123210 nnn

b

a

n xfxfxfxfxfxfxfx

Sdxxf ++++++∆

=≈ −−

where n is even and n

abx

−=∆

10. Use the trapezoidal rule to find an approximation

to 2

1

2

dxe x

using 4=n .

11. Use Simpson’s rule to find an approximation to

2

1

2

dxe x

using 4=n .

12.


10

13.

a) Assume ( )f x is continuous, approximate 3

0( )f x dx using LRAM and RRAM and TRAP

b) Approximate (1)f ′

14.

a) Assume ( )f x is continuous, approximate 8

1( )f x dx using LRAM and RRAM and TRAP

b) Approximate (7)f ′


11

15. So, we now have methods for approximating

definite integrals, but the question still

remains…how good are these approximations???

As an example, let’s consider an integral that we can

evaluate exactly: 4

1

2

1dx

x. The exact value is

32 .

Here are the associated errors:

Error Bounds

If ET and EM are the errors in the Trapezoidal and

Midpoint rules, respectively, then

2

3

12

)(

n

abKET

−≤ 2

3

24

)(

n

abKEM

−≤

Where Kxf ≤)('' for bxa ≤≤

For Simpson’s Rule, 4

5

180

)(

n

abKES

−≤

where K is now such that Kxf ≤)()4(

16.

Suppose we approximate 3

1

1dx

x using Midpoint

Rule with 5=n

a) What is the maximum possible error?

b) If we want a maximum possible error

of 8

10−

, what value of n should we

use?

17.

If we approximate 2

0

cos

π

xdx using Simpson’s

Rule with 4=n , what is the maximum possible

error?


12

ASSIGNMENT Sigma Limits of Finite Sums (AP) Review gr11 math:

1. 4 7 10 13___... 52.− − − − −

a) Number of terms

b) Sum

c) Sigma notation

2. 32 16 8 4 ___ ...+ + + + +

a) Term formula

b) Sum formula

c) Sigma notation

Simplify the following sums

3.

4.

5.

6. Definite integral of f from a to b is

1

( ) lim ( )nb

ka n

k

f x dx f x x→∞

=

= ∆

For continuous functions f , where the interval

],[ ba is divided into n subintervals of equal width

b ax

n

−∆ = and

kx a k x= + ∆

Ex. Express 2

1

5lim

cos

n

kk

nk k

xx x

x→∞=

+ ∆

as a

definite integral on

4,0π

.


13

7. Areas Under Curve versus Integrals

Cannot simply evaluate the definite integral! Area

is always ______________! So when using area to

find definite integrals, we are responsible for

assigning the regions the correct ___________.

This means you must find where the graph

_________________________________ then split

up our interval, manually making negative regions

positive: ( ) ( )

c b

a c

f x dx f x dx+

If you’re using a calculator just enter:

Note: placement of absolute value matters!

8. Set up two separate integral expressions that

would give the actual area of the region bounded

by the function 2

( ) 4f x x= − and the x-axis on

the interval [1,5]

9.

Find

4

3

1

( 2 )x x dx− using sums. Does the

answer represent area?

10. Find the area under

2)( xxf = between 0=x

and 1=x using sums.


14

11.

Find −

−

4

2

)63( dxx using sums

check your answer graphically using geometric

area formulas

12.

Find +

3

1

)25( dxx using geometric area

13.

Find

5

1

3x dx−

− using geometric area


15

ASSIGNMENT Definite Integrals (AP)

1. Just like

0

( ) ( )lim

h

f x h f x

h→

+ − was defined to be the

___________________________

the 1

lim ( )n

kn

k

f x x→∞

=

∆ is defined to be the

___________________________.

Definite integral of f from a to b is

1

( ) lim ( )nb

ka n

k

f x dx f x x→∞

=

= ∆

For continuous functions f , divide the interval ],[ ba into n

subintervals of equal width b a

xn

−∆ = and

kx a k x= + ∆

ie. 0 1 2

( ), , , , ( )n

x a x x x b= =…

Integrability of Continuous Functions

Theorem

If a function f is continuous over the

interval [a, b], then the definite integral

( )b

af x dx

exists and f in integrable over [a, b]

Actually the theorem is also true for

f that has at most ______________

many _______________ discontinuities

Integration Properties

If f and g are integrable on given intervals

1. −=a

b

b

adxxfdxxf )()( ,

2. =a

adxxf 0)( ,

3. −=b

aabccdx )( , c is any constant

4. [ ]( ) ( ) ( ) ( )b b b

a a af x g x dx f x dx g x dx± = ±

5. =b

a

b

adxxfcdxxcf )()( , c is any

constant

6. =+c

a

b

a

c

bdxxfdxxfdxxf )()()(

2. Given the following information:

7)(

8

2

= dxxf , 2)(

8

2

−= dxxg , 9)(

10

2

= dxxf

evaluate these definite integrals:

a) =+ dxxgxf

8

2

)](4)(2[

b) =+ dxxg ]5)(6[

8

2

c) = dxxf

10

8

)(

d)

2

8

( )f x dx

−

−

=

3. 4 6

0 4

6 4

0 6

4 4

0 6

( ) 8 and ( ) 2

a . ( ) b. ( )

c. 4 ( ) d. ( )

f x dx f x dx

f x dx f x dx

f x dx f x dx

−

−

= =

= =

− = =


16

4.

5.


17

6.

7. Suppose we want to approximate the area under

( ) 2f x x= − between 0x = and 6x = .

a) Find an expression for this area as a limit.

b) Do you know how to evaluate this sum?

c) Rewrite as a definite integral on [0, 6], why does this

no longer represent area?

d) Do you know how to find the indefinite integral for

this function?

8. Suppose we want to approximate the area under

xxf sin)( = between 0=x and 2

π=x .

a) Find an expression for this area as a limit.

b) Do you know how to evaluate this sum?

c) Rewrite this limit as a definite integral

d) Do you know how to find the indefinite integral for

this function?

9. For the above questions we need the __________________________________________to connect the concepts c) and d)…Without it

we can learn more tedious sum simplifications or use a finite n to approximate the answer


18

ASSIGNMENT FTC & Average Value of a Function (AP)

1. Fundamental Theorem of Calculus (FTC): Suppose

f is continuous on ],[ ba . Define F as:

p1. ( ) ( ) , x

aF x f t dt a x b= ≤ ≤

then F is continuous on [a,b] and differentiable

on (a,b), and ( ) ( ) ( )x

a

dF x f t dt f x

dx′ = =

p2. −=b

aaFbFdxxf )()()( , where F is

any antiderivative of f .

First part states that __________________ of

________________________ gives back the

function. Note: ___________ limit must be x.

Second part helps you to evaluate definite integrals

without _______________________, it also gives

you _____________________________

Idea behind part 2

Proof of part 1

Notation explanation of part 1


19

Review indefinite integrals

2.

++−+ dx

xexx

x2

76sin4

3

3. 2

2

17 1 csc

(2 5) 2

xx dx

x

+ + +

−

Practice FTC part 2

4.

( )2

3

12

xdx

−

5. 36

1( 3)

xe dx−

6. 1

2

1

3 5dx

x− −

7. 1

20

2

64 16

dx

x−

8. 2

0(sin 3 1)x dx

π

+

9. 3

2

0

sec 5 x dx

π

10. Does this method for evaluating definite integrals

always work? Consider the following example

dxxdxx

−

−

−

=

1

1

2

1

1

2

1

11.


20

12.

−2

1)( dxxf where

≥+

<

=1,

11

1,2

)(x

x

xx

xf

13.

+3

0)1( dxxx

14. Use Sums, what other method(s) would work?

15. Use Geometric Shapes, what other method(s) would work?

− +1

3)23( dxx

16. Use FTC part 2, what other method(s) would work?

17. 3

21

2

1

xdx

x+ see the need for substitution

method to be learned in the next unit


21

Find area bounded by: Find area bounded by:

18.

19.

20. Use the symmetry to find the area bounded by x-

axis and the given function on [-1, 1]

21. Find the average value of )sin()( xxf = on [ ]π,0

22.

dxx

xx−

+

−5

5

2

7

1

43 use symmetry too

23. The temperature of a 5m long metal rod is given byxexf 4

)(−= at a distance of x metres from one end

of the rod. What is the average temperature of the

rod?


22

ASSIGNMENT More FTC & MVT for Integrals (AP)

1.

2.

3. Find the interval on which the curve

3 2

0( 1)

x

y t t dt= + + is concave up. Justify your

answer.

4.


23

Practice FTC part 1

5.

x

dtttdx

d

3

2sin

6.

7. 03

1x

dt dt

dx+

8.

−

173

x tdt

te

t

dx

d

9.

10.

11.

12.

+3

5

21

x

xdtt

dx

d


24

13. The Mean Value Theorem for Integrals: If f is

continuous on ],[ ba , then there exists a number

c in ],[ ba such that

))(()( abcfdxxf

b

a

−=

Demonstrate why theorem would break down for

non continuous functions.

14. Find the value of c guaranteed by the MVT for

integrals for 2

( ) 1f x x= + on [-1,2]. Interpret the

result graphically.


derivatives for 2

( ) 1f x x= + on [-1,2]. Interpret

the result graphically.


integrals for

17. Find the equation of the tangent line to

2

1( ) cos

x

g x tdt= at ( ,2)π


25

The Net Change Theorem: The integral of a rate of

change is the net change:

−=′b

aaFbFdxxF )()()(

Or Accumulation:

what I have now = what I started with + what I've accumulated since the start

( ) ( ) ( )b

aF b F a F x dx′= +

24. If water flows from a tank at a rate of

ttr 2100)( −= litres per minute for 500 ≤≤ t ,

find the amount of water that flows from the tank

in the first 10 minutes.

25. During 34 weeks in the 2000-2001 flu season, the

rate of reported influenza per 100,000 people in

Ireland could be approximated by tetI 1049.0

389.3)( =′ , where I is the total number of

people per 100,000 who have contracted influenza

and t is time measured in weeks. Approximately

how many people per 100,000 contracted

influenza during the whole 34 weeks?

26. At the start of Christmas Break, at t = 0 days, a man

weighed 180 pounds. If the man gained weight

during the break at a rate modeled by the function

( ) 10sin8

tW t

π ′ =

pounds per day, what was

the man’s weight (in pounds) at the end of the

break, 14 days later?

27. If ( ) 4cos

6

xf x

π ′ =

and

3(2)f

π=

a) Find ( 2)f − use symmetry

b) Find ( )f x


26

Solve for the indicated variable

28.

29.

30. Evaluate

10

2

0

6 x dx−

31. Find the number(s) b such that the average value

of 2

( ) 2 6 3f x x x= + − on the interval [0,b] is

equal to 3.

32. Suppose that

0

( )3 2

xf t

dt xt

+ = Find f(x)


27

33.

34. Population is given by ( ) ln(3 1)P t t= + thousand

of insects where t is number of years since Jan

2000.

a) Find average population from t=5 to t=9

b) Find average change in population from

t=5 to t=9


28

ASSIGNMENT Interpret the Integral (AP)

1.

The graph represents a person out for a walk. The

person was 2m to the left of an oak tree at t=0 sec.

a) What direction is the person travelling? When

did she turn around?

b) What does the integral represent?

c) Find the position at t=10sec

d) Find the average velocity on [0,10] using both

formulas given in the journal

e) Find average acceleration on [0,10]

2. If )(th′ is the rate of change of a child’s height

measured in inches per year, what does the

integral ′10

0

)( dtth represent, and what are its

units?

3. Population is growing with a rate

100 25dP

tdt

= + people/year where t is time

since 2000 in years.

a) Record an integral to be used to predict

the population in 2010 if in 2000

population was 30 000 people.

b) Can you solve this integral with techniques

we know so far?

4.

Suppose 5.6mg of dye was injected into the

bloodstream.

a) Record an integral that would represent

the area of under this curve. What does it

represent?

b) Find the efficiency with which the dye is

being processed in L/sec.


29

5.


30

6.

(d) Find g(-4)


31

7.


32

8.

(d) Sketch the displacement time graph


33

9. The average value of homes in a valley is growing

at a rate of 4t + thousand dollars per year

where t represents the number of years since

2005. If the average value of a home in this area

was $90 000 in 2007 determine when the average

value of a home will be $110 000

10. If oil is leaking from a ruptured tanker at a rate of 0.2

( ) 10tf t e= gallons/hr where t is measured in

hours since the tanker was damaged. Evaluate and

explain the meaning of:

a) 5

0( )f t dt

b) 12

0

1( )

12f t dt

11. A particle moves in a straight line so that its

velocity at time t is given by tttv 2)(2 −= m/s.

a) Find the displacement of the particle

during the first 3 seconds.

b) Find the distance travelled by the particle

during the first 3 seconds.

12. WITH CALC The rate at which people enter an amusement park on any

given day is 2

15600( )

24 160E t

t t=

− + people/hr, and the

rate at which people are leaving is 2

9890L( )

38 370t

t t=

− +

where t is measured in hours after midnight. The park is open

from 9am to 11pm.

a) How many people have entered the park by 5pm?

b) How many people are at the park at 5pm?

c) How does the number of people at the park

changing at 5pm?

Documents

Big idea Feedback & Assessment of Your Successabout the function itself? ex. you know velocity function but what to know the _____ function. A function F is called an antiderivative