welcome to the integral drill and practice power point flash drill! developed by susan cantey at...
TRANSCRIPT
Welcome to the Integral Drill and Practice Power Point Flash Drill!
Developed by Susan Cantey
at Walnut Hills H.S.
2006
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.