CALCULUS II

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Go Back To Calculus I

Review The Definite Integral

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Exponential And Log Functions

Trigs And Inverse Trigs

Hyperbolics

Integration By Substitution

Integration By Parts

Integration By Partial Fractions

Integration By Trig Substitution

Areas And Volumes Of Solids

l'Hospital's Rule

Improper Integrals

The Transcendental Functions

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Graph of f(x)=x separating f(x)=eþ (above) and f(x)=ln(x).
Exponential Function:
Log Function: or or
Natural Log Function:

Exponential Derivatives

Example:

General Form:

Ex:

Logarithmic Derivatives

Ex
Ex
Ex
Ex

 

Exponential Integrals

General:

Logarithmic Integration

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Trig And Inverse Trig Functions

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Hyperbolic Functions


 
y=sinh-1(x) if and only if x=sinh(y)

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Integration By Substitution

1) This is the chain rule in reverse.
2)Inspect the integrand for the appearance of f(x) and f'(x), then state:
u=f(x) 
<u=f'(x) <x, then substitute.

Example:

u=1+x2

<u=2x <x or<u=x <x

Substitute: 

Now substitute the original expression:

Linear Substitutions


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Integration By Parts/Tabular Integration

This method is useful for some products, logarithms and inverse trigonometric functions.

The fomula may be expressed in two ways:

#f(x)*g'(x) <x=f(x)*g(x)-#f'(x)*g(x) <x
#u <v=uv-#v <u

Notice that one part is chosen to integrate and the other part is chosen to differentiate.

Example:

#x sin(x) <x

u=x <v= sin(x) <x

<u=(1) <x v=-cos(x) 

-x cos(x)-#-cos(x) <x

-x cos(x)+sin(x)

For more information on Integration by parts, click Here.

Trigonometric Reduction Formulas

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Integration By Partial Fractions

Example:

Substitute x to find A abd B, or multiply this equation out.

A=3, B=5

Now, Integrate.

For denominators like:

Use: 

For denominators like: 

Use:

For more information on integration by partial fractions, click Here.

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Integration By Trigonometric Substitution

1)

let 

2) 

let 

3)

let 

Example:

2#(16-x2)1/2 <x

The form is: a2-x2

a=4

x=4 sin q

<x=4 cos q <q

(16-x2)1/2=4 cos q

2#(4 cos q)(4 cos q) <q

32#cos2 q <q

=16[(sin q)(cos q)+q] 

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Areas And Volumes Of Solids

Area Between Graphs

Between f(x) andg(x).
Between g(y) and y axis.
Between function of y curves.

Volumes

(General form). (General form).
For disk method. Rotates around x axis. 
For disk method. Rotates around y axis.
Remember by: For disk method.
For washer method.
Remember by: For Washer method.
Area:
For shell method. r=x, h=f(x)

Arc Length

Parametric: x=f(t); y=g(t)

Surface Areas

Rotates around x axis.
Rotates around y axis.
Remember by:
Rotates around x axis.
Rotates around y axis.
Remember by:
Parametric:  Rotates around y axis.
Rotates around x axis.
Remember by:  r=distance to the axis.
Polar:  Rotates around y axis.
Rotates around x axis.
Remember by:  Rotates around y axis.
Rotates around x axis.

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l'Hospital's Rule


 


Some limits are hard to evaluate directly. When you try, you may end up with an Indeterminate Form like one of these seven types: 0/0, \/\, 0*\, 00, \0, 1\, \-\.

In the above, \ may be replaced by -\. Ex. 0*(-\) is indeterminate but \+\ isn't.

l'Hospital's Rule lets us find limits of indeterminate forms of the first two types. The rest are handled by converting them to one of the first two (creating a quotient) and then applying l'Hospital's Rule.

BE CERTAIN you are dealing with an indeterminate form before applying the Rule or it won't work. l'Hospital's Rule allows us to calculate lim [f'(x)/g'(x)] instead of lim [f(x)/g(x)] under certain conditions.

1) Case 0/0 where

If 

,

where L is finite, \, or -\, then

.

2) Case \/\ where

The conclusion is the same 

as for above (Case 0/0).

Sometimes, you need to apply the Rule more than once in a problem. The rule also works for left-hand and right-hand limits and for limits as x approaches \ or -\.

Take This has the form 0*(-\). 
We can't apply l'Hospital's Rule directly, so we re-write it as:

Now the form is .

Now apply l'Hospital's Rule:

In dealing with the three indeterminate forms with exponents, (00, \0, 1\), We need to use logarithms.

has the form 1\.
So what does lny approach as x approaches \
has the form \*0, so rewrite it to 0/0 form:
Apply l'Hospital's Rule: 
Remember: you're NOT applying the quotient rule!
So 
e3 is the answer.

See more thrilling examples of each case for l'Hospital's Rule Here!

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Improper Integrals


 


For a continuous function x>or=a, the improper integral is defined:

For x>=0, think of the area under the curve to the right of x=a.

1) If the limit exists and is finite, the integral converges.

2) If the limit is not finite or doesnt exist, the integral diverges.

So the area under the curve is infinite.

Notice that for this function, the volume of the solid of rotation is finite!

Now suppose that f is continuous for x<or=b. The improper integral is:

The integral is the area under the curve.

We Define:

Only if both integrals on the right converge. If either integral on the right diverges, the integral on the left diverges.

An example of a converging function over + or - infinity

A vertical asymptote occurs at x=b in this example:

If the asymptote is at x=a, you get:

Although it is not being evaluated at infinity for x, the following is another example of an improper integral:

Solving using integration by parts gives a value of approx. -.6137.

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