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Limits
Definitions
Precise Definition : We say lim ( ) x a
f x L Æ
= if
for every e > 0 there is a d > 0 such that
whenever 0 < x - a < d then f (^) ( x (^) )- L < e.
“Working” Definition : We say lim ( ) x a
f x L Æ
if we can make f (^) ( x (^) )as close to L as we want
by taking x sufficiently close to a (on either side
of a ) without letting x = a.
Right hand limit : lim ( ) x a
f x L Æ +
=. This has
the same definition as the limit except it
requires x > a.
Left hand limit : lim ( ) x a
f x L Æ-
=. This has the
same definition as the limit except it requires
x < a.
Limit at Infinity : We say lim ( ) x
f x L Æ•
= if we
can make f (^) ( x ) as close to L as we want by
taking x large enough and positive.
There is a similar definition for lim ( ) x
f x L Æ-•
except we require x large and negative.
Infinite Limit : We say lim ( ) x a
f x Æ
= • if we
can make f (^) ( x ) arbitrarily large (and positive)
by taking x sufficiently close to a (on either side of a ) without letting x = a.
There is a similar definition for lim ( ) x a
f x Æ
except we make f (^) ( x (^) )arbitrarily large and
negative.
Relationship between the limit and one-sided limits
lim ( ) x a
f x L Æ
= fi lim (^) ( ) lim ( ) x a x a
f x f x L Æ +^ Æ-
= = lim (^) ( ) lim ( ) x a x a
f x f x L Æ +^ Æ-
= = fi lim ( ) x a
f x L Æ
lim (^) ( ) lim ( ) x a x a
f x f x Æ +^ Æ-
π fi lim ( ) x a
f x Æ
Does Not Exist
Properties
Assume lim ( ) x a
f x Æ
and lim ( ) x a
g x Æ
both exist and c is any number then,
- lim (^) ( ) lim ( ) x a x a
cf x c f x Æ Æ
È ˘ =
Î ˚
- lim ( ) ( ) lim ( ) lim ( ) x a x a x a
f x g x f x g x Æ Æ Æ
ÈÎ ± ˘˚= ±
- lim (^) ( ) ( ) lim (^) ( ) lim ( ) x a x a x a
f x g x f x g x Æ Æ Æ
È ˘ =
Î ˚
( )
( )
( )
( )
lim lim lim
x a x a x a
f x^ f^ x
g x g x
Æ Æ Æ
È ˘
Í ˙=
Î ˚
provided lim ( ) 0 x a
g x Æ
π
- lim (^) ( ) lim (^) ( )
n n
x a x a
f x f x Æ Æ
È ˘ =^ È^ ˘
Î ˚ Î ˚
- lim n^ ( ) n lim ( ) x a x a
f x f x Æ Æ
È ˘ =
Î ˚
Basic Limit Evaluations at ± •
Note : (^) sgn ( a )= 1 if a > 0 and (^) sgn ( a )= - 1 if a < 0.
- lim
x x Æ•
e = • & lim 0
x x Æ- •
e =
- lim ln( ) x
x Æ•
= • & (^) ( ) 0
lim ln x
x Æ -
- If r > 0 then lim (^) r 0 x
b Æ• (^) x
- If r > 0 and
r x is real for negative x
then lim 0 x r
b
Æ-• (^) x
- n even : lim
n x
x Ʊ •
- n odd : lim n x
x Æ•
= • & lim n x
x Æ- •
- n even : lim sgn( )
n x
a x b x c a Ʊ •
+ L+ + = •
- n odd : lim sgn( ) n x
a x b x c a Æ•
+ L+ + = •
- n odd : lim sgn( )
n x
a x c x d a Æ-•
+ L+ + = - •
Evaluation Techniques Continuous Functions
If f (^) ( x (^) )is continuous at a then lim ( ) ( ) x a
f x f a Æ
Continuous Functions and Composition
f ( x )is continuous at^ b^ and^ lim ( ) x a
g x b Æ
= then
lim ( ( )) ( lim ( )) ( )
x a x a
f g x f g x f b Æ Æ
Factor and Cancel
( ) ( )
( )
2
2 2 2
2
lim lim 2 2
6 8 lim 4 2
x x
x
x x^ x^ x
x x x x
x
x
Æ Æ
Æ
+ - -^ +
Rationalize Numerator/Denominator
( )( ) (^ )^ ( )
( ) ( )
9 2 9 2
9 2 9
lim lim 81 81 3
9 1 lim lim 81 3 9 3
x x
x x
x x x
x x (^) x
x
x x x x
Æ Æ
Æ Æ
Combine Rational Expressions
( )
( )
( ) ( )
0 0
0 0 2
lim lim
lim lim
h h
h h
x x h
h x h x h x x h
h
h x x h x x h x
Æ Æ
Æ Æ
Ê ˆ^ Ê^ -^ + ˆ
Á -^ ˜ =^ Á^ ˜
+ Á^ + ˜
Ë ¯ Ë ¯
Ê - ˆ -
= Á ˜= = -
Á + ˜ +
Ë ¯
L’Hospital’s Rule
If
( )
( )
lim x a 0
f x
Æ (^) g x
= or
( )
( )
lim x a
f x
Æ g x
then,
( )
( )
( )
( )
lim lim x a x a
f x f x
Æ (^) g x Æ g x
a is a number, • or -•
Polynomials at Infinity
p x ( ) and q x ( )are polynomials. To compute
( )
( )
lim x
p x
Ʊ• q x
factor largest power of x out of both
p x ( ) and^ q x ( )and then compute limit.
2 2
(^2 )
(^2 )
4 4
5 5
lim lim lim x (^) 5 2 x (^) x 2 x (^) x 2 2
x (^) x
x x Æ-• (^) x x Æ- • (^) x Æ-•
Piecewise Function
( ) 2
lim x
g x Æ-
where ( )
2 5 if 2
1 3 if 2
x x g x x x
Ï + < -
= Ì
Ó-^ ≥ -
Compute two one sided limits,
( )
2 2 2
lim lim 5 9 x x
g x x Æ- -^ Æ--
( ) 2 2
lim lim 1 3 7 x x
g x x Æ- +^ Æ-+
One sided limits are different so (^) ( ) 2
lim x
g x Æ- doesn’t exist. If the two one sided limits had
been equal then (^) ( ) 2
lim x
g x Æ-
would have existed
and had the same value.
Some Continuous Functions Partial list of continuous functions and the values of x for which they are continuous.
Polynomials for all x.
Rational function, except for x ’s that give division by zero.
n x ( n odd) for all x.
n x ( n even) for all x ≥ 0.
x e for all x.
- ln x for x > 0.
- cos ( x ) and sin ( x )for all x.
- tan ( x )and sec( x )provided
3 3 , , , , , 2 2 2 2
x
p p p p π L - - L
- cot ( x )and csc( x )provided
x π L , - 2 p , - p , 0, p , 2 p ,L
Intermediate Value Theorem
Suppose that f (^) ( x (^) )is continuous on [ a, b ] and let M be any number between f (^) ( a ) and f (^) ( b (^) ).
Then there exists a number c such that a < c < b and f (^) ( c (^) )= M.
Derivatives
Definition and Notation
If y = f ( x )then the derivative is defined to be ( )
( ) ( ) 0
lim h
f x h f x f x Æ h
If (^) y = f ( x )then all of the following are
equivalent notations for the derivative.
df dy d f x y f x Df x dx dx dx
If (^) y = f ( x )all of the following are equivalent
notations for derivative evaluated at x = a.
( ) ( ) x a x a x a
df dy f a y Df a = (^) dx (^) = dx =
Interpretation of the Derivative
If y = f (^) ( x )then,
- m = f ¢( a )is the slope of the tangent
line to y = f (^) ( x )at x = a and the
equation of the tangent line at x = a is
given by y = f (^) ( a ) (^) + f ¢( a (^) )( x - a ).
- f ¢ (^) ( a )is the instantaneous rate of
change of f (^) ( x (^) )at x = a.
- If f (^) ( x (^) )is the position of an object at
time x then f ¢( a )is the velocity of
the object at x = a.
Basic Properties and Formulas
If f (^) ( x (^) )and g (^) ( x ) are differentiable functions (the derivative exists), c and n are any real numbers,
- (^) ( c f (^) ) ¢^ = c f ¢( x )
- (^) ( f ± g (^) ) ¢^ = f ¢ (^) ( x ) (^) ± g ¢( x )
- (^) ( f g (^) )¢^ = f ¢ g + f g ¢ – Product Rule
2
f f g f g
g g
Ê ˆ ¢ - ¢
Á ˜=
Ë ¯
- Quotient Rule 5. ( ) 0
d c dx
d (^) n n 1 x n x dx
= – Power Rule
d f g x f g x g x dx
= ¢^ ¢
This is the Chain Rule
Common Derivatives
( ) 1
d x dx
( sin^ ) cos
d x x dx
( cos^ ) sin
d x x dx
( )
2 tan sec
d x x dx
( sec^ ) sec^ tan
d x x x dx
( csc^ ) csc^ cot
d x x x dx
( )
2 cot csc
d x x dx
1 2
sin 1
d x dx (^) x
1 2
cos 1
d x dx (^) x
1 2
tan 1
d x dx x
( ) ln( )
d (^) x x a a a dx
d x x
dx
e = e
ln , 0
d x x dx x
ln , 0
d x x dx x
= π
log , 0 ln
a
d x x dx x a
Chain Rule Variants The chain rule applied to some specific functions.
d n^ n^1 f x n f x f x dx
È ˘ = È ˘^ ¢ Î ˚ Î ˚
( )
( ) (^ )^
d (^) f x f x ( ) f x dx
e = ¢ e
( )
( )
ln
d f x f x dx f x
È ˘=
Î ˚
4. ( sin ( )) ( ) cos ( )
d f x f x f x dx
È ˘ = ¢ È ˘
Î ˚ Î ˚
5. ( cos ( )) ( ) sin ( )
d f x f x f x dx
È ˘ = - ¢ È ˘
Î ˚ Î ˚
2 tan sec
d f x f x f x dx
È ˘ = ¢ È ˘
Î ˚ Î ˚
7. ( sec [ f ( ) x ] ) f ( ) x sec [ f ( ) x ] tan[ f ( ) x ]
d
dx
( )
( )
1 2 tan 1
d f x f x dx (^) f x
È ˘ =
Î ˚
+ È ˘
Î ˚
Higher Order Derivatives The Second Derivative is denoted as
( )
( ) ( )
2 2 2
d f f x f x dx
¢¢ (^) = = and is defined as
f ( x ) ( f ( x ) )
¢¢ = ¢ , i.e. the derivative of the
first derivative, f ¢(^ x ).
The n
th Derivative is denoted as
( ) ( )
n n n
d f f x dx
= and is defined as
( ) ( )
( )
( (^ ))
n n 1 f x f x
= , i.e. the derivative of
the ( n -1) st derivative, ( ) ( )
n 1 f x
Implicit Differentiation
Find y ¢ if (^) ( ) 2 9 3 2 sin 11
x y x y y x
e + = +. Remember^ y = y ( x )here, so products/quotients of^ x^ and^ y
will use the product/quotient rule and derivatives of y will use the chain rule. The “trick” is to differentiate as normal and every time you differentiate a y you tack on a y ¢ (from the chain rule).
After differentiating solve for y ¢^.
( ) ( )
( )
( )
2 9 2 2 3 2 9 2 2 2 9 2 9 2 2 3 3 2 9 3 2 9 2 9 2 2
2 9 3 2 cos 11 11 2 3 2 9 3 2 cos 11 2 9 cos 2 9 cos 11 2 3
x y x y x y x y x y x y x y
y x y x y y y y x y y x y x y y y y y x y y x y y y x y
- ¢^ + + ¢^ = ¢+
e e e e e e e
Increasing/Decreasing – Concave Up/Concave Down Critical Points
x = c is a critical point of f (^) ( x ) provided either
1. f ¢ (^) ( c ) = 0 or 2. f ¢( c )doesn’t exist.
Increasing/Decreasing
- If f ¢ (^) ( x ) > 0 for all x in an interval I then
f (^) ( x (^) )is increasing on the interval I.
- If f ¢ (^) ( x ) < 0 for all x in an interval I then
f (^) ( x (^) )is decreasing on the interval I.
- If f ¢ ( x ) = 0 for all x in an interval I then
f ( x )is constant on the interval I.
Concave Up/Concave Down
- If f ¢¢( x ) > 0 for all x in an interval I then
f (^) ( x ) is concave up on the interval I.
- If f ¢¢( x ) < 0 for all x in an interval I then
f (^) ( x ) is concave down on the interval I.
Inflection Points
x = c is a inflection point of f (^) ( x ) if the
concavity changes at x = c.
Integrals
Definitions
Definite Integral: Suppose f (^) ( x (^) )is continuous
on (^) [ a b ,]. Divide (^) [ a b ,] into n subintervals of
width D x and choose
x i from each interval.
Then (^) ( ) ( )
1
lim (^) i
b
a (^) n i
f x dx f x x Æ• (^) =
= (^) Â D Ú
Anti-Derivative : An anti-derivative of f (^) ( x )
is a function, F (^) ( x ) , such that F ¢( x (^) ) = f (^) ( x ).
Indefinite Integral : f ( x dx ) = F ( x )+ c Ú
where F (^) ( x (^) )is an anti-derivative of f (^) ( x (^) ).
Fundamental Theorem of Calculus
Part I : If f (^) ( x ) is continuous on (^) [ a b ,] then
( ) ( )
x
a
g x = f t dt Ú is also continuous on (^) [ a b ,]
and (^) ( ) ( ) ( )
x
a
d g x f t dt f x dx
Ú
Part II : f (^) ( x (^) )is continuous on[ a b ,] , F (^) ( x (^) )is
an anti-derivative of f (^) ( x (^) )( i.e. F (^) ( x (^) ) = f (^) ( x dx ) Ú
then (^) ( ) ( ) ( )
b
a
f x dx = F b - F a Ú
Variants of Part I :
( )
( ) ( ) ( )
u x
a
d f t dt u x f u x dx
= ¢ È ˘
Ú Î ˚
( ) ( )
( ) ( )
b
v x
d f t dt v x f v x dx
= - ¢ È ˘
Ú Î ˚
( ) ( )
( ) ( ) [ ( )^ ] ( ) [ ( )]
u x
v x
u x v x
d f t dt u x f v x f dx
Ú
Properties
Ú f^^ (^ x )^^ ±^ g x dx (^ )^^ =^ Ú f^ (^ x dx )^^ ±Ú g^ (^ x dx )
( ) ( ) ( ) ( )
b b b
a a a Ú^ f^ x^ ±^ g^ x dx^ =^ Ú f^ x dx^ ±Ú g^ x dx
( ) 0
a
a Ú f^ x dx^ =
( ) ( )
b a
a b Ú^ f^ x dx^ = -Ú f^ x dx
Ú cf^^ (^ x dx )^^ = c^ Ú f^ (^ x dx ) ,^ c^ is a constant
( ) ( )
b b
a a Ú^ cf^ x dx^ = c^ Ú f^ x dx ,^ c^ is a constant
( ) ( )
b b
a a Ú^ f^ x dx^ =Ú f^ t dt
( ) ( )
b b
a a
f x dx £ f x dx Ú Ú
If f (^) ( x (^) ) ≥ g (^) ( x )on a £ x £ b then (^) ( ) ( )
b a
a b
f x dx ≥ g x dx Ú Ú
If f (^) ( x (^) ) ≥ 0 on a £ x £ b then (^) ( ) 0
b
a
f x dx ≥ Ú
If m £ f (^) ( x ) £ M on a £ x £ b then (^) ( ) ( ) ( )
b
a
m b - a £ f x dx £ M b - a Ú
Common Integrals
k dx = k x + c Ú 1 1 1
n n n
x dx x c n
= + π - Ú (^1 ) x dx (^) xdx ln x c
Ú =^ Ú =^ + 1 1 a ln a x b
dx ax b c
Ú
Ú ln^ u du^ =^ u^ ln^ (^ u^ )-^ u^ + c u u du = + c Ú e e
cos u du = sin u + c Ú
sin u du = - cos u + c Ú 2 Ú sec^ u du^ =^ tan u^ + c
sec u tan u du = sec u + c Ú
csc u cot udu = - csc u + c Ú 2 Ú^ csc^ u du^ = -^ cot u^ + c
tan u du = ln sec u + c Ú
sec u du = ln sec u + tan u + c Ú
( )
1 1 1 2 2 tan^
u a u du^ a^ a c
Ú =^ +
( )
1 2 2
1 sin u a a u
du c
Ú
Standard Integration Techniques Note that at many schools all but the Substitution Rule tend to be taught in a Calculus II class.
u Substitution : The substitution u = g (^) ( x )will convert (^) ( ( )) ( ) ( ) ( )
b g b ( )
a g a
f g x g ¢ x dx = f u du Ú Ú using
du = g ¢ ( x dx ). For indefinite integrals drop the limits of integration.
Ex. (^) ( ) 2 3 2
1
5 x cos x dx Ú (^3 2 2 ) u = x fi du = 3 x dx fi x dx = 3 du 3 3 x = 1 fi u = 1 = 1 :: x = 2 fi u = 2 = 8
( ) (^ )
( ) (^) ( ( ) ( ))
(^2 2 38 )
1 13
5 8 5 3 1 3
5 cos cos
sin sin 8 sin 1
x x dx u dx
u
Ú Ú
Integration by Parts : u dv = uv - v du Ú Ú and
b (^) b b
a a a
u dv = uv - v du Ú Ú
. Choose u and dv from
integral and compute du by differentiating u and compute v using v = (^) Ú dv.
Ex.
x x dx
Ú e x x u x dv du dx v
= = e fi = = - e x x x x x x dx x dx x c
Ú e^^ = -^ e^ +^ Ú e^ = -^ e^ -^ e +
Ex.
5
3 Ú ln^ x dx
1 u = ln x dv = dx fi du = (^) xdx v = x
( ( ) )
( ) ( )
(^5 55 )
3 3 3 3
ln ln ln
5ln 5 3ln 3 2
x dx = x x - dx = x x - x
Ú Ú
Products and (some) Quotients of Trig Functions
For sin cos n m x x dx Ú we have the following :
1. n odd. Strip 1 sine out and convert rest to
cosines using
2 2 sin x = 1 - cos x , then use the substitution u = cos x.
2. m odd. Strip 1 cosine out and convert rest
to sines using 2 2 cos x = 1 - sin x , then use the substitution u = sin x.
3. n and m both odd. Use either 1. or 2. 4. n and m both even. Use double angle and/or half angle formulas to reduce the integral into a form that can be integrated.
For tan sec n m x x dx Ú we have the following :
1. n odd. Strip 1 tangent and 1 secant out and convert the rest to secants using 2 2 tan x = sec x - 1 , then use the substitution u = sec x. 2. m even. Strip 2 secants out and convert rest
to tangents using 2 2 sec x = 1 + tan x , then use the substitution u = tan x.
3. n odd and m even. Use either 1. or 2. 4. n even and m odd. Each integral will be dealt with differently.
Trig Formulas : sin 2( x ) = 2sin ( x ) cos( x ), ( ) ( ( ))
(^2 ) cos x = 21 + cos 2 x , ( ) ( ( ))
(^2 ) sin x = 21 - cos 2 x
Ex.
3 5 tan x sec x dx Ú
( )
( ) ( )
3 5 2 4
2 4
2 4
1 7 1 5 7 5
tan sec tan sec tan sec
sec 1 sec tan sec
1 sec
sec sec
x xdx x x x xdx
x x x xdx
u u du u x
x x c
Ú Ú
Ú
Ú
Ex.
5 3
sin cos
x x Ú dx
( )
1 2 1 2 2 2
5 4 2 2 3 3 3 2 2 3 (^2 2 2 ) 3 3
sin sin sin (sin )^ sin cos cos cos (1 cos )sin cos (1 ) (^) 1 2
cos
sec 2 ln cos cos
x x x x^ x x x x x x x u (^) u u u u
dx dx dx
dx u x
du du
x x x c
Ú Ú Ú
Ú
Ú Ú
Trig Substitutions : If the integral contains the following root use the given substitution and
formula to convert into an integral involving trig functions.
2 2 2 sin a b a - b x fi x = q
2 2 cos q = 1 - sin q
2 2 2 sec a b b x - a fi x = q
2 2 tan q = sec q - 1
2 2 2 tan a b a + b x fi x = q
2 2 sec q = 1 +tan q
Ex. (^) 2 2
16 x (^) 4 9 x
dx
Ú
2 2 3 3 x = sin q fi dx = cos q dq
2 2 2 4 - 9 x =^4 -^ 4 sin^^ q^ =^ 4 cos^ q^ =2 cos q
Recall 2 x = x. Because we have an indefinite
integral we’ll assume positive and drop absolute
value bars. If we had a definite integral we’d
need to compute q ’s and remove absolute value
bars based on that and,
if 0
if 0
x x x x x
Ï ≥
= Ì
Ó-^ <
In this case we have 2 4 - 9 x =2 cos q.
( )
( )
2 3 sin 2cos 2
4 2 2 9
16 12 sin
cos
12 csc 12 cot
d d
d c
q q^ q
q q q
q q
Û
ı Ú
Ú Use Right Triangle Trig to go back to x ’s. From
substitution we have 3 2 sin x q = so,
From this we see that 4 9^2 3 cot x x q
=. So,
2 (^2 )
16 4 4 9 4 9
x x x x
dx c
Ú
Partial Fractions : If integrating
( ) ( )
P x Q x
dx Ú where the degree of P ( x )is smaller than the degree of
Q x ( (^) ). Factor denominator as completely as possible and find the partial fraction decomposition of
the rational expression. Integrate the partial fraction decomposition (P.F.D.). For each factor in the
denominator we get term(s) in the decomposition according to the following table.
Factor in Q x ( (^) ) Term in P.F.D Factor in Q x ( ) (^) Term in P.F.D
ax + b
A
ax + b
( )
k ax + b ( ) ( )
1 2 2
k k
A A A
ax b (^) ax b ax b
L
2 ax + bx + c 2
Ax B
ax bx c
( )
2 k ax + bx + c ( )
1 1 (^2 )
k k k
A x B A x B
ax bx c (^) ax bx c
L
Ex. (^) 2 ( )( )
2 1 4
7 13 x x
x x dx
Ú
( ) (^ )
2 2
2 2
( )( )
3 2 1 2 2
(^2 43 ) 1 4 1 4 4 3 16 (^1 4 )
7 13
4 ln 1 ln 4 8 tan
x x x x x x x (^) x x
x x
x
dx dx
dx
x x
= +
Ú Ú
Ú
Here is partial fraction form and recombined.
2 2 2 2
- ( ) ( ) ( )( ) ( )( )
(^2 ) 1 4 1 4 1 4
7 13 ( Bx^ C^ x x x x x x x
x x A Bx C^ A x +^ +^ +^ -
= + =
Set numerators equal and collect like terms.
( ) ( )
2 2 7 x + 13 x = A + B x + C - B x + 4 A - C
Set coefficients equal to get a system and solve to get constants. 7 13 4 0
4 3 16
A B C B A C
A B C
An alternate method that sometimes works to find constants. Start with setting numerators equal in
previous example : (^) ( ) ( ) ( ) 2 2 7 x + 13 x = A x + 4 + Bx + C x - 1. Chose nice values of x and plug in.
For example if x = 1 we get 20 = 5 A which gives A = 4. This won’t always work easily.
Applications of Integrals
Net Area : ( )
b
a Ú f^ x dx represents the net area between^ f^ (^ x^ )and the
x -axis with area above x -axis positive and area below x -axis negative.
Area Between Curves : The general formulas for the two main cases for each are,
( ) upper function lower function
b
a
y = f x fi A = (^) Ú ÈÎ^ ˘˚^ - ÈÎ^ ˘˚ dx & ( ) right function left function
d
c
x = f y fi A = (^) Ú ÈÎ^ ˘˚^ - ÈÎ^ ˘˚ dy
If the curves intersect then the area of each portion must be found individually. Here are some sketches of a couple possible situations and formulas for a couple of possible cases.
( ) ( )
b
a
A = f x - g x dx Ú
( ) ( )
d
c
A = (^) Ú f y - g y dy ( ) ( ) ( ) ( )
c b
a c
A = f x - g x dx + g x - f x dx Ú Ú
Volumes of Revolution : The two main formulas are V = A x dx ( ) Ú and V = A ( (^) y dy ) Ú
. Here is
some general information about each method of computing and some examples. Rings Cylinders
( (^ )^ (^ ))
2 2 A = p outer radius - inner radius A = 2 p ( radius ) ( width / height)
Limits: x / y of right/bot ring to x / y of left/top ring Limits : x / y of inner cyl. to x / y of outer cyl.
Horz. Axis use f (^) ( x ) ,
g (^) ( x ) , A x ( (^) )and dx.
Vert. Axis use f (^) ( y (^) ),
g (^) ( y (^) ), A y ( (^) )and dy.
Horz. Axis use f (^) ( y ) ,
g (^) ( y ) , A y ( (^) )and dy.
Vert. Axis use f (^) ( x (^) ),
g (^) ( x (^) ), A x ( (^) )and dx.
Ex. Axis : y = a > 0 Ex. Axis : y = a £ 0 Ex. Axis : y = a > 0 Ex. Axis : y = a £ 0
outer radius : a - f (^) ( x )
inner radius : a - g ( x )
outer radius: a + g (^) ( x )
inner radius: a + f ( x )
radius : a - y
width : f (^) ( y ) (^) - g (^) ( y )
radius : a + y
width : f ( y ) - g ( y )
These are only a few cases for horizontal axis of rotation. If axis of rotation is the x -axis use the
y = a £ 0 case with a = 0. For vertical axis of rotation ( x = a > 0 and x = a £ 0 ) interchange x and
y to get appropriate formulas.
