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

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?

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) • = 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]

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!

The indefinite integral = ?

Well…hard to write; easy to say The indefinite integral equals the general antiderivative… = 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 APexam….

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

= ?

ax + C I hope you got that one!

= ?

+ C Ready?

= ??

- cos x + C Don’t forget we are going backwards! So if the derivative was positive, the anti-derivative is negative.

sin x + C Got the negative/positive situation straight?? Good!

= ???

OK that’s a hard one! ln|tanx+sec x|+C If you got it right, you deserve a little treat!

= ?

tan x + C That should have been easy! Piece of cake! Upside down!!

= ??

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

=??

ln |x| +C You need the absolute value in case x<0 Rise to the highest! Sursum ad Summum yada yada

where n > 1 Hint:

1/xn = x-n sooooooo……. the answer is: + C You didn’t say ln(xn) did ya??

= ?

ex + c Easiest anti-derivative in the universe, eh?

= ?

sec x + C Another easy peasy as a daisy anti-derivative!

= ?

Not toooo difficult? -cot x + C Safe landing?

= ??

-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!

= ???

+ C Grin and bear it!! Ha Ha

OK! Take a deep breath! 5 more questions!

tan-1x + C Keep it going!!

sin-1x + C Oh yeah! Only 3 more to go.

sec-1x + C It’s all down hill now!!!!

(Did you get the significance of the picture?)

R U ready 4 the last ? ?

= ???

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