This document provides an overview of anti-derivatives and integrals. It explains that anti-derivatives are used to find the original function given information about how that function changes, and that they can be used to build models of real-world phenomena like electron flow in circuits. The document also discusses how integrals allow us to find the value of a function over an interval rather than indefinitely, and introduces double and triple integrals for 2D and 3D spaces.

1. WHO USES CALCULUS
ANYWAYS?
CALCULUS

2. DONEC QUIS NUNC

3. ANTI-DERIVATIVES

4. OK WHAT?
ANTI-DERIVATIVES
• So anti-derivatives are like starting at the end of a
movie and rewinding it to the beginning.
• With a derivative - you see how the function changes
over time
• With an anti-derivative - you’re given how the function
changes, but want the original function.

5. LET’S BREAK THIS DOWN
ANTI-DERIVATIVES
Take the
derivative
Of speed
This gives you
acceleration
As your distance changes with time, you get speed
What does it mean to change you distance?
d
dt
(speed) = acceleration

6. LET’S BREAK THIS DOWN
ANTI-DERIVATIVES
Take the
antiderivative
Of
acceleration
This gives
you Speed
As your distance changes with time, you get speed
What does it mean to change you distance?
∫
(acceleration)dt = speed

7. LET’S BREAK THIS DOWN
ANTI-DERIVATIVES
Take the
antiderivative
Of
acceleration
This gives
you Speed
As your distance changes with time, you get speed
What does it mean to change you distance?
∫
(speed)dt = position

8. Acceleration
ANTI-DERIVATIVES
Position
Speed
Take a derivative
Take a derivative Take a anti-derivative
Take a anti-derivative

9. LET’S LOOK AT A FEW EXAMPLES OF WHAT THE ANTI-DERIVATIVE LOOKS LIKE
ANTI-DERIVATIVE EXAMPLES
∫
6tdt
∫
2tdt
∫
3t2
dt
∫
(speed)dt
∫
5t6
dt
∫
7t8
dt
Remember, the dt at the end is just something we put there!
It doesn’t really do anything.

10. 3 is one more than 2.
We can take the anti-
derivative!
∫
3t2
dt
One rule - if the number in front of the
t is one more than the power
Then - copy and paste the number in
front of the t and paste it in the power
(up top and to the right of the t)

11. 4 is one more than 3.
We can take the anti-
derivative!
∫
4t3
dt
One rule - if the number in front of the
t is one more than the power
Then - copy and paste the number in
front of the t and paste it in the power
(up top and to the right of the t)

12. 2 is NOT one more
than 5. We cannot
take the anti-
derivative!
∫
2t5
dt
One rule - if the number in front of the
t is one more than the power
Then - copy and paste the number in
front of the t and paste it in the power
(up top and to the right of the t)

13. HOW TO FIND THE ANTI-DERIVATIVE
ANTI-DERIVATIVE EXAMPLES
∫
7t6
dt = ?
Remember, the dt at the end is just something we put there!
It doesn’t really do anything.
∫
3t2
dt = ?
∫
20t19
dt = ?
One rule - if the number in front of the
t is one more than the power
Then - copy and paste the number in
front of the t and paste it in the power
(up top and to the right of the t)

14. HOW TO FIND THE ANTI-DERIVATIVE
ANTI-DERIVATIVE EXAMPLES
∫
4t3
dt = ?
Remember, the dt at the end is just something we put there!
It doesn’t really do anything.
∫
3t2
dt = t3
∫
4t5
dt = Can′t do this yet!
One rule - if the number in front of the
t is one more than the power
Then - copy and paste the number in
front of the t and paste it in the power
(up top and to the right of the t)

15. LET’S BUILD A MODEL
• Well, you guys and girls have been
doing this all along.
• Let’s say we want to model the
movement of an electrons over time.
• How do we measure how something
changes?
• What do we need to build the
model?
• We will start with an equation!

16. MODEL BUILDING
• What do we call the movement of electrons along a conductor?
•
• is current, and represents the electrons.
• What does the mean?
• This is the scientific model for electricity!
I =
dQ
dt
I Q
d
dt

17. MODEL BUILDING
• Remember the original function describes the amount of charge, .
• When we take the derivative, we get current.
• or
• What would happen if we took the anti-derivative of current.
• You got it, we would get back the original function
Q
i =
dQ
dt
current =
d
dt
(the number of electrons)

18. WHAT’S WRONG WITH OUR MODEL?
• What about if the electrons bump up and down
• What if there is a kink or bend in the wire
• What if there are a lot of protons around that attract the electrons away from the
current?
• What is the battery is dying?

19. A LITTLE MORE ABOUT ANTI-DERIVATIVES
• When we take an anti-derivative, we do it like this:
•
• When we take an anti-derivative, we get a function that spans all of time up to
infinity.
• This is like asking a friend how their day was and 6 hours later they are still telling
you about when they were five and ate a pb&j with the crust off.
• Sometimes the function is too much information, because you can put any time in
and it will give you some information.
∫
idt = Q

20. A LITTLE MORE ABOUT ANTI-DERIVATIVES
•
• When we take an anti-derivative, we get back a function.
• What if we take an anti-derivative of
• Yes, that is .
• Well, we have only really seen a fraction of what this function looks like:
∫
idt = Q
∫
2tdt = ?
t2

21. 30
300 40000
Not so bad. Doesn’t go that far out
Ok, ok, its going places. Still no too impressive:|

22. 80,000,000 3,000,000,000
Um what, that is 7 zeros! Ok, we get the point. The function
goes on forever and never ever ever
ever ever ever ever ever ever stops.

23. ANTI-DERIVATIVES
• Sometimes we do not want a function that goes on forever and ever and ever and
ever and ever and…
• Sometimes we just want to know what is happening between two times.
• Like how many electrons are there between 0 seconds and 20 seconds?
• Well, anti-derivatives can this!
•
, these are called boundaries for the function.
• This will give you back only the time you put into the anti-derivative.
∫
t2
t1
idt

24. ANOTHER MATH TERM
• When you go look anti-derivatives up:
• They are also called integrals
•
• Just know that, I didn’t want to confuse you with some many terms.
∫
dt = antiderivative = integral

25. COOL INTEGRALS
• The integrals we have worked with are just for 1-d examples.
• Like walking in a straight line ————————————
• But sometimes we need to describe things in 2d and 3d. These are called double
and triple integrals.
• (double integral)
• (triple integral)
∫ ∫
f(x, y, z) dx dy
∫ ∫ ∫
f(x, y, z) dx dy dz

26. COOL INTEGRALS
• Cylinders have there own special integrals
•
• Spheres like a beach ball have there own integrals as well
•
∫ ∫ ∫
f(ρ, ϕ, z)ρ dρ dϕ dz
∫ ∫ ∫
f(r, θ, ϕ)r2
sinθ dr dθ dϕ