Graph of f(x)=x separating f(x)=eþ (above) and f(x)=ln(x).

Exponential Function:

Log Function: or or

Natural Log Function:

Example:

Ex:

	Ex:
	Ex:
	Ex:
	Ex:

General:

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

u=1+x2 ‹u=2x ‹x or ‹u=x ‹x Substitute: Now substitute the original expression:

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:

1) let

2) let

3) let

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]

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

(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)

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

	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.

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.

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:

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.

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:

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: