Math 54 Reviewer problems

U P K E M

M E M B E R S H I P A C A D E M I C D E V E L O P M E N T

Math 53

4th Long Exam Notes

Heavily based from

C.G. Tapia lectures

Preface

This handout is intended as a reviewer only and

should not be substituted for a complete lecture,

or used as a reference material. The goal of this

reviewer is to refresh the student on the concepts

and techniques in one reading. But this is more

than enough to replace your own notes :)

1 Antiderivatives or Indefinite

Integrals

Theorem 1. If F is an antiderivative of f on

an interval I, then every antiderivative of f on

I is given by F (x) + C, where C is an arbitrary

constant.

Remark 1.1. The antiderivative of F is not

unique, and F1 and F2 differ only by a constant.

1.1 Theorems on Antidifferentiaion

1.

dx = x+ C

2. If a is any constant, thenaf(x) dx = a inf f(x) dx

3. If f and g are defined on the same interval,(f(x) g(x)) dx =

f(x) dx

g(x) dx

4. If n is any rational number and n 6= 1, thenxn dx =

xn+1

n+ 1+ C

1.2 Antiderivatives of Trigonometric Func-

tions

1.

sinx dx = cosx+ C

2.

cosx dx = sinx+ C

3.

sec2 x dx = tanx+ C

4.

csc2 x dx = cotx+ C

5.

secx tanxdx = secx+ C

6.

cscx cotxdx = cscx+ C

1.3 Substitution Rule

1

University of the Philippines Chemical Engineering Society, Inc. (UP KEM)

Math 53 4th Long Exam Reviewer

Theorem 2. If u = g(x) is a differentiable func-

tion whose range is an interval I and f is con-

tinuous on I, thenf(g(x))g(x) dx =

f(u) du

1.4 Rectilinear Motion Revisited

Suppose that a particle is traveling along a straight

line and s(t), v(t), anda(t) are the equations of mo-

tion, velocity and acceleration, respectively, of the

particle. Also,

v(t) = s(t) and a(t) = v(t)

Therefore, s(t) is a particular antiderivative of

v(t) while v(t) is a particular antiderivative of

a(t).

2 Definite Integral

2.1 Area of a Plane Region.

Riemann Sum

Definition (Summation notation). If n is a posit-

ive integer and F is a function such that {1, 2, . . . , n}is in the domain of F , then

ni=1

F (i) := F (1) + F (2) + + F (n)

2.1.1 Some summation identities

If n is a positive integer, c R and F and G arefunctions defined on the set {1, 2, . . . , n},

1.

ni=1

c = cn

2.

ni=1

cF (i) = c

ni=1

F (i)

3.

ni=1

(F (i) +G(i)) =

ni=1

F (i) +

ni=1

G(i)

4.

ni=1

i =n(n+ 1)

2

5.

ni=1

i2 =n(n+ 1)(2n+ 1)

6

2.1.2 Area of a Plane Region

Definition (Riemann Sum Limit). The area of a

plane region AR is given by the limit of a Riemann

sum:

AR = limn

ni=1

f(xi ) x

where xi is any number in the ith subinterval, theheight of the ith rectangle is f(xi ), and x is thewidth of the ith rectangle

2.2 The Definite Integral

Definition (Definite Integral). Let f be defined

on [a, b]. The definite integral of f from a to b is ba

f(x) dx = limxk0

ni=1

f(xi ) xk

If the limit exists and does not depend on the

choice of partitions or on the choice of numbers

xi in the subintervals, the function is said to beintegrable on [a, b].

Remark. xk is the width of a partition under

a curve and f(xi ) as the height of the partition.Also, if a function is continuous on [a, b], then it

is integrable on [a, b].

2.3 Properties of the Definite Integral

Let f and g be integrable on [a, b], and let c R.

1.

ba

f(x) dx = ab

f(x) dx

2.

aa

f(x) dx = 0

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3.

ba

cdx = c(b a)

4.

ba

cf(x) dx = c

ba

f(x) dx

5. ba

(f(x) g(x)) dx = ba

f(x) dx ba

g(x) dx

6. If f is integrable on a closed interval I containing

{a, b, c}, ba

f(x) dx =

ca

f(x) dx+

bc

f(x) dx

Regardless of the order of a, b, c.

3 The Fundamental Theorem

of Calculus

Theorem 3. If the functions f and g are integ-

rable on [a, b], and if f(x) g(x) for all x in [a, b],then b

a

f(x) dx ba

g(x) dx

Theorem 4. Suppose f is continuous on the

closed interval [a, b]. If m and M are the absolute

minimum function value and absolute maximum

function value, respectively, of f in [a, b], then

m(b a) ba

f(x) dx M(b a)

Theorem 5 (Mean Value Theorem for Integrals).

If the function f is continuous on the closed in-

terval [a, b], then there exists a number c in [a, b]

such that ba

f(x) dx = f(c)(b a)

Definition (Average value). If the function f is

integrable on [a, b], the average value of f on [a, b]

is

fave =

ba

f(x) dx

b a

3.0.1 The First Fundamental Theorem of Cal-

culus

Theorem 6. Let f be a function continuous on

[a, b] and let x be any number in [a, b]. If F is the

function defined by

F (x) =

xa

f(t) dt

then

F (x) = f(x)

Remark 6.1. Suppose

F (x) =

g(x)a

f(t) dt

where f is a function continuous on [a, b] and let

g(x) [a, b]. If we let

H(x) =

xa

f(t) dt

then F (x) = H(g(x)).

Using the chain rule, we get

F (x) = H (g(x))g(x)

By the First Fundamental Theorem of Calculus

H (x) = f(x). So, F (x) = f(g(x))g(x).

3.0.2 The Second Fundamental Theorem of Cal-

culus

Theorem 7. Let f be a function continuous on

[a, b]. If F is any antiderivative of f on [a, b], then

ba

f(x) dx = F (x)

x=bx=a

= F (b) F (a)

Remark 7.1. By the Second Fundamental The-

orem of Calculus and the Substitution Rule, ba

f(g(x))g(x) dx = g(b)g(a)

f(u) du

letting u = g(x)

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4 Applications of the Definite

Integral

4.1 Area of a Plane Region

Theorem 8. If f and g are continuous functions

on the interval [a, b] and f(x) g(x) for all x [a, b], then the area of the region R bounded above

by y = f(x), below by y = g(x) and the vertical

lines x = a and x = b is

AR =

ba

(f(x) g(x)) dx

where [a, b] is the interval I covered by the region

along the x-axis.

Remark 8.1. When horizontal rectangles are

used, the variable of integration is y and the in-

tegrand is expressed in terms of y.

4.2 Arc Length

Theorem 9. If y = f(x) is a smooth curve on the

interval [a, b], then the arc length L of this curve

from x = a to x = b is

LR =

ba

1 +

(dy

dx

)2dx =

ba

1 + [f (x)]2 dx

where [a, b] is the interval I covered by the region

along the x-axis.

4.3 Solids of Revolution

4.3.1 Discs or Washers

1. Disc method (that is, the solid is not hollow).

Suppose we have a region R bounded above by

r(x), below by the y = y0 and on the sides

by x = a and x b. The volume of the solidgenerated by revolving R about y = y0 is:

VR = pi

ba

([r(x)]2 y20

)dx

Remark. For disc method, r(x) is orthogonal

to the variable of integration. In this particular

case, we use vertical discs. The formula can be

reworked to horizontal discs. When y0 = 0, we

revolve the region about the x-axis. If y0 >

r(x) for all x [a, b], then switch them to geta positive value.

2. Washers (that is, the solid has hollow parts

or a boundary of R does not lie on the axis

of revolution). Suppose we have a region R

bounded above by r2(x), below by r1(x) and

on the sides by x = a and x b. The volumeof the solid generated by revolving R about

y = y0 6= r1(x) 6= r2(x) for some x [a, b]is:

VR = pi

ba

([r2(x)]

2 [r1(x)]2)

dx

Remark. Almost similar remarks as for the

disc method. Just keep in mind that r2(x) is

always the one farther away from the axis of

revolution.

4.3.2 Cylindrical shells

Suppose R is the region bounded above by y =

f(x), below by y = g(x), and the vertical lines

x = a and x = b and let f and g be continuous

functions on [a, b]. If the line x = x0 does not

intersect the interior of R, then the volume of the

solid of revolution obtained when R is revolved

about the line x = x0 is given by:

VR = 2pi

ba

r(x)h(x) dx

Where r(x) and h(x) are the radius and the

height, respectively, of a cylindrical shell at an

arbitrary x in [a, b].

Remark. The particular case above can be re-

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worked for horizontal shells.

Remark. In all cases above, we considered R to

have left and right bounds x = a and x = b. In

some cases, R is completely bounded by just the

two functions and {(a, f(x)), (b, f(b)} are the in-tersection points of the said functions.

4.4 Volume by Slicing

Let S be a solid bounded by two parallel planes

perpendicular to the x-axis at x = a and x =

b. If the cross-sectional area of S in the plane

perpendicular the x-axis at an arbitrary x in [a, b]

is given by a continuous function A(x), then the

volume of the solid is

V =

ba

A(x) dx

Page 5 of 5

Antiderivatives or Indefinite IntegralsTheorems on AntidifferentiaionAntiderivatives of Trigonometric FunctionsSubstitution RuleRectilinear Motion Revisited

Definite IntegralArea of a Plane Region.Riemann SumSome summation identitiesArea of a Plane Region

The Definite IntegralProperties of the Definite Integral

The Fundamental Theorem of CalculusThe First Fundamental Theorem of CalculusThe Second Fundamental Theorem of Calculus

Applications of the Definite IntegralArea of a Plane RegionArc LengthSolids of RevolutionDiscs or WashersCylindrical shells

Volume by Slicing

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Math 54 Reviewer problems