Download Course: AP Calculus BC - D. Caldwell

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
a
f ( x)dx
Here’s a few facts:

b
f ( x)dx returns the
1. If f(x) > 0, then
a
numerical value of the area between
f(x) and the x-axis (area “under” the curve)
2.

b
f ( x)dx = F(b) – F(a) where F(x) is
a
any anti-derivative of f(x).
(Fundamental Theorem of Calculus)
3. Basically

b
a
f ( x)dx gives the total cumulative
change in f(x) over the interval [a,b]
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.
n
 f (x
k 1
k
) x k
=
area of n rectangles
As n approaches infinity… x  dx
n
and  f ( x k )  b f ( x)
k 1

a
So the definite integral sums infinitely
many infinitely thin rectangles!
The indefinite integral
f
(
x
)
dx
=
?

Well…hard to write; easy to say
The indefinite integral equals the general
antiderivative…
f
(
x
)
dx

= F(x) + C
Where F’(x) = f(x)
Now let’s see if you’ve memorized specific
anti-derivatives that you will need to know quickly
during the AP exam….

1 x
dx
sin x  tan x
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!
x
dx

n
=?
1
n 1
Ready?
x
n 1
+C
sin
xdx

= ??
- cos x + C
Don’t forget we are
going backwards!
So if the derivative was
positive, the
anti-derivative is
negative.
cos
xdx

=?
sin x + C
Got the negative/positive situation straight??
Good!
sec
xdx

= ???
OK that’s a hard one!
ln|tanx+sec x|+C
If you got it right, you deserve a
little treat!
sec
xdx

2
=?
tan x + C
That should have been easy!
Piece of cake! Upside down!!
tan
xdx

= ??
If you forget this one
think: “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
1
dx
x
=??
ln |x| +C
You need the absolute value in case x<0
Rise to the highest! Sursum ad Summum
yada yada

Hint:
1
dx
n
x
where n > 1
1/xn = x-n
sooooooo…….
the answer is:
1
 n 1
x
 n 1
+C
You didn’t say ln(xn) did ya??
e
dx

x
=?
x
e +
c
Easiest anti-derivative in the universe, eh?
sec
x
tan
xdx

=?
sec x + C
Another easy peasy as a daisy anti-derivative!
csc
xdx

2
=?
Not toooo difficult?
-cot x + C
Safe landing?
csc
x
cot
xdx

= ??
-csc x + C
How are you holding up?
Bored out of your gourd?
Suck it up! You’ll thank me when you test out of
college calculus!
a
dx

x
= ???
1
ln a
x
a + C
Grin and bear it!! Ha Ha
OK! Take a
deep breath!
5 more
questions!
1
dx
 1 x2
?
-1
tan x
+C
Keep it
going!!

1
1 x
?
2
dx
-1
sin x
Oh yeah! Only 3
more to go.
+C
| x|
1
x 1
2
?
dx
-1
sec x
+C
It’s all down hill now!!!!
udv

?

udv

uv

vdu


(Did you get the significance of the picture?)
R U ready
4 the last ?
?

1
dx
( x  a)( x  b)
= ???

1
B 
 A
dx   

dx
( x  a)( x  b)
 xa xb
= A ln(x-a) + B ln(x-b) + C
(I’m assuming you know how to find A & B)
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.