A moment of silence for our great calculus “father” please.

OK…here we go!

Integrals: Drill & Practice

• I’m going to ask you about integrals.

• It’s important to be as fast as possible because time is your enemy .

• When you think you know the answer,

(or if you give up ) click to get to the next slide to see if you were correct.

First let’s talk about what the integral means!

Can you list some interpretations of the definite integral?

b

adxxf )(

Here’s a few facts:

1. If f(x) > 0, then returns the numerical value of the area between f(x) and the x-axis (area “under” the curve)

2. = F(b) – F(a) where F(x) is any anti-derivative of f(x). (Fundamental Theorem of Calculus)

3. Basically gives the total cumulative

change in f(x) over the interval [a,b]

b

adxxf )(

b

adxxf )(

b

adxxf )(

What is a Riemann Sum?

Hint: Here’s a picture!

A Riemann sum is the area of n rectangles used to approximate the definite integral.

= area of n rectangles

As n approaches infinity…

and

So the definite integral sums infinitely many infinitely thin rectangles!

n

kkk xxf

1

)(

dxx

b

a

n

kk xfxf )()(

1

The indefinite integral

= ? dxxf )(

Well…hard to write; easy to say

The indefinite integral equals the general antiderivative…

= F(x) + C Where F’(x) = f(x) dxxf )(

Now let’s see if you’ve memorized specific anti-derivatives that you will need to know quickly

during the AP exam….

dxxx

x

tansin

1 2

sike!

I just made that one up to scare you…now the rest will seem easy!

= ?adx

ax + C

I hope you got that one!

= ?

dxx n

+ C

Ready?

111

nn x

= ??

xdxsin

- cos x + C

Don’t forget we are going backwards!

So if the derivative was positive, the

anti-derivative is negative.

=? xdxcos

sin x + CGot the negative/positive situation straight??

Good!

= ???

xdxsec

OK that’s a hard one!

ln|tanx+sec x|+CIf you got it right, you deserve a

little treat!

= ?

xdx2sec

tan x + CThat should have been easy!

Piece of cake! Upside down!!

= ?? xdxtan

If you forget this onethink: “tan x = sin x / cos x”

(then let u = cos x, du = - sin x dx, etc.)

- ln(cos x) + C

or

ln(sec x) + C

=??

dxx

1

ln |x| +CYou need the absolute value in case x<0

Rise to the highest! Sursum ad Summum

yada yada

where n > 1

Hint:

dxx n1

1/xn = x-n

sooooooo…….the answer is:

+ C

You didn’t say ln(xn) did ya??

11

1

nn x

= ?

dxe x

ex + cEasiest anti-derivative in the universe, eh?

= ?

xdxx tansec

sec x + C

Another easy peasy as a daisy anti-derivative!

= ?

xdx2csc

Not toooo difficult?

-cot x + C

Safe landing?

= ??

xdxx cotcsc

-csc x + CHow are you holding up?

Bored out of your gourd?Suck it up! You’ll thank me when you test out of

college calculus!

= ???

dxa x

+ C

Grin and bear it!! Ha Ha

xa aln1

OK! Take a deep breath!

5 more questions!

?

dx

x 21

1

tan-1x + C

Keep it going!!

?

dxx 21

1

sin-1x + COh yeah! Only 3

more to go.

?

dx

xx 1||

12

sec-1x + C

It’s all down hill now!!!!

?udv

(Did you get the significance of the picture?)

vduuvudv

R U ready4 the last ?

?

= ???

dx

bxax ))((

1

= A ln(x-a) + B ln(x-b) + C

(I’m assuming you know how to find A & B)

dxbx

B

ax

Adx

bxax

))((

1

You’re done!Ta Ta for now.

Be sure to check out these other power point slide shows:

Derivatives

Pre-Calculus Topics (on a separate page)

Sequences and Series

Miscellaneous Topics

and

Additional BC Topics

I said you are done!

Stop clicking.