Multivector Review and Training Center

INTEGRAL CALCULUS

Integration Formulas
1.  a du  a  du  au  C , where a = constant
u n 1
u du   C , where n ≠ -1
n
2.
n1
du
3. 
u
 ln u du  C

4.  ln u du  u ln u  u  C

e du  eu  C
u
5.
au
a du  C
u
6.
ln a
7.  u dv  uv   v du (Integration by Parts)
8.  sin u du   cos u  C
9.  cos u du  sin u  C
10.  tan u du  ln sec u  C   ln cos u  C
11.  cot u du  ln sin u  C   ln csc u  C
12.  sec u du  ln sec u  tan u  C
13.  csc u du  ln csc u  cot u  C   ln csc u  cot u  C
 sec u du  tan u  C
2
14.

 csc u du   cot u  C
2
15.
16.  sec u tan u du  sec u  C
17.  csc u cot u du   csc u  C
du u
18.   arc sin
a
C
a u
2 2

du 1 u
19.  u2  a2 
a
arc tan  C
a
du 1 u
20.  
a
arc sec  C
a
u u a 2 2

21.  sinh u du  cosh u  C

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22.  cosh u du  sinh u  C

 sech u du  tanh u  C
2
23.

 csc h u du   coth u  C
2
24.
25.  sech u tanh u du   sech u  C
26.  csc h u coth u du   csc h u  C
du 1 u a
27.  u2  a2  ln    C , if u2 > a2
2 a  u  a 
du 1 u  a
28.  a2  u2  ln    C , if u2 < a2
2 a  u  a 

 ln  u  u 2  a 2   C
du
29.   
u a
2 2

 ln  u  u 2  a 2   C
du
30.   
a u
2 2

u 2 a2 u
31.  a2  u2 
2
a  u2 
2
arc sin  C
a
a2 
ln  u  u 2  a 2   C
u 2
32.  u2  a2 
2
u  a2 
2  

Walli’s Formula

2  m  1m  32 or 1n  1n  32 or 1 

0 sin2  cos n  d  
m  nm  n  2m  n  42 or 1
 
 

where: m and m are non-negative integers (0, 1, 2, 3, 4, ……)

α = π/2, if both m and n are even, or one is zero and the other is even
α = 1, if otherwise

Note: If m and n equals 1 or 0, apply the following:

Rule: If the factor (m – 1) or (n – 1) in the numerator is 0 or –1, replace

the product in which this occurs by unity.

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Plane Areas in Rectangular Coordinates System

(b, d)
y = f(x)
R

x
0 (a, c) y = g(x)
y

I. Using a vertical rectangular element (vertical stripping)

y y = f(x) y = g(x)
dA  y u  y L  dx (b, d)

A
b
a
y u  y L dx dA yu - yL
yu

A   f ( x)  g( x) dx
b dx yL
a (a, c) x
0

where: U – upper
L – lower

II. Using a horizontal rectangular element (horizontal stripping)

y y = f(x) or
x = f(y) y = g(x) or
(b, d) x = g(y)
dA  x R  x L  dy xL

x R  x L dy
d dA dy
A
c
(a, c)
A
d
c
g(y )  f (y ) dy 0
x
xR - xL

xR

Using Polar Coordinates

1 2 2
2  1
A r d
r
d

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Volume of Solid of Revolution

y
y

x
x
0

Torus

Methods of Finding the Volumes of Solid of Revolution

I. Disk Method

Rules: 1. The axis of rotation is a part of the boundary of the plane area.
2. The element chosen must be parallel to the axis of rotation.

dh

r
Axis of dV
h=a dh h=b Rotation r

dV   r 2 dh
b
V    r 2 dh
a

II. Ring or Washer Method

Rules: 1. The axis of rotation is not a part of the boundary of the plane area.
2. The element chosen must be parallel to the axis of rotation.

dh

r2 r2
r1 r1
r1
h=a h=b r2

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
dV   r22   r12 dh 

dV   r22  r12 dh
b
 
V   r22  r12 dh
a

III. Cylindrical Shell Method

Rules: 1. The axis of rotation may or may not be a part of the boundary of
the rotated area.
2. The element chosen must be parallel to the axis of rotation.

dr

r = a dr r = b
r
dV  2rh dr
b
V  2  r h dr
a

Pappus Theorem

First Proposition
The surface area of revolution is equal to the length of the generating arc times the
circumference of the circle described by the centroid of the arc, provided the axis
of revolution does not cross the generating arc.

B
A
A s  CL c
As  2  c L 0
x

where: L – length of the arc

Second Proposition
The volume of the solid revolution is equal to the generating area times the
circumference of the circle described by the centroid of the area, provided the axis
of revolution does not cross the generating arc.

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y
V  AC  A 2  c 
V  2 c A c
x
Centroid of a Plane Area

A x   xc dA
A y   y c dA

where: xc and yc are coordinates of the centroid of rectangular element

x and y are coordinates of the centroid of area A

y
(x, y)
xc  x
1 (xc, yc)
y
yc  y yc
2 x
0 xc

y
x
1
xc  x
2
yc  y (x, y)
yc
x
0
xc

y
xL
yc  y xc
x  xL dy
xc  R  xL (xc, yc)
2 yc = y
x  xL
xc  R x
2
xR

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x
xc
xc  x
yu  yL
yc   yL yu - yL
yu
2
yc
y  yL
yc  u yL
x
2

Centroid of a Solid of Revolution

V x   xc dV
V y   y c dV
SHELL
dV DISK
WASHER

Area of Common Polar Curves

r  2asin , A  a 2 r 2  a 2 cos , A  2a 2

r  2acos , A  a 2

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r 2  a 2 cos 2, A  a 2 r 2  a 2 sin 2, A  a 2

a 2 3a 2
r  asin 3, A  r  a1  sin  , A 
12 2

1 2 1 2
r  acos 2, A  a r  asin 2, A  a
2 2

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