Warmup 11/30/15
How was your Thanksgiving? What are you most thankful
for in your life?
Objective
Tonight’s Homework
To define the definite integral
pp 251: 3, 4, 5, 6, 9, 11
Homework Help
Let’s spend the first 10 minutes of class going over any
problems with which you need help.
The Fundamental Theorem of Calculus I
We’re far enough along now to define the main,
fundamental equation for calculus.
The Fundamental Theorem of Calculus I
We’re far enough along now to define the main,
fundamental equation for calculus.
We’ve seen that we can calculate
the area under a curve as a series
of infinite rectangles.
The Fundamental Theorem of Calculus I
We’re far enough along now to define the main,
fundamental equation for calculus.
We’ve seen that we can calculate
the area under a curve as a series
of infinite rectangles.
Take a minute and do the following:
Find the area under y = x from 0 to 5.
Then find the integral of y = x. Plug in 5 for x
after doing this and tell me what you get.
The Fundamental Theorem of Calculus I
The Fundamental Theorem of Calculus
If a function f(x) has an antiderivative ( F(x) )
over the range from a to b, then…
b
∫a f(x) dx = F(b) – F(a)
The Fundamental Theorem of Calculus I
The Fundamental Theorem of Calculus
If a function f(x) has an antiderivative ( F(x) )
over the range from a to b, then…
b
∫a f(x) dx = F(b) – F(a)
We call the above operation a “definite integral”
because we evaluate it with numbers to get an
end result.
Why is this so amazing? Because this value is
also the area under the curve.
The Fundamental Theorem of Calculus I
Example:
Find the area under f(x) = sin(x) from 0 to π.
The Fundamental Theorem of Calculus I
Example:
Find the area under f(x) = sin(x) from 0 to π.
b
π
∫a f(x) dx
[-cos(x)]
1+1
∫0sin(x) dx
π
-cos(π) - -cos(0)
0
2
The Fundamental Theorem of Calculus I
There’s another thing to note here. The definite
integral can differentiate between area above
the x-axis and area below the x-axis.
The Fundamental Theorem of Calculus I
There’s another thing to note here. The definite
integral can differentiate between area above
the x-axis and area below the x-axis.
2π
Example:
Calculate
∫0 sin(x)
The Fundamental Theorem of Calculus I
There’s another thing to note here. The definite
integral can differentiate between area above
the x-axis and area below the x-axis.
2π
Example:
Calculate
2π
[-cos(x)]
-cos(2π) - - cos(0)
0
-1 + 1
∫0 sin(x)
0
The Fundamental Theorem of Calculus I
There’s another thing to note here. The definite
integral can differentiate between area above
the x-axis and area below the x-axis.
2π
Example:
Calculate
2π
[-cos(x)]
-cos(2π) - - cos(0)
0
-1 + 1
∫0 sin(x)
0
How do we get 0 area?!
The Fundamental Theorem of Calculus I
From 0 to π, our function is above zero, but
from π to 2π, our function spends just as much
time below zero.
These two portions cancel out. The first half is
positive and the second half is negative.
The Fundamental Theorem of Calculus I
One last idea for today.
We defined this area under the curve initially as
a sum of rectangles.
But which rectangles?
We used lower sums, but we could have used
upper sums or even midpoint sums.
The Fundamental Theorem of Calculus I
In the mid 1800s, a mathematician named
Riemann realized that this was a problem. He
wanted to see a generic way of getting sums.
The Fundamental Theorem of Calculus I
In the mid 1800s, a mathematician named
Riemann realized that this was a problem. He
wanted to see a generic way of getting sums.
With this in mind, he created the Riemann sum.
This is a summing up of rectangles like we’ve
seen before, but now,
each rectangle can be
any width and can touch
the function at any
point, even if every
rectangle is different.
The Fundamental Theorem of Calculus I
He summed this up with the following function:
Riemann Sum = lim
||P|| 0
i=n
∑
i=1
f(xi)Δxi
Where ||P|| is the width of the widest partition.
The Fundamental Theorem of Calculus I
He summed this up with the following function:
i=n
Riemann Sum = lim
||P|| 0
∑
i=1
f(xi)Δxi
Where ||P|| is the width of the widest partition.
2
Example: Write
∫0 e
x
dx as a Riemann sum
The Fundamental Theorem of Calculus I
He summed this up with the following function:
i=n
∑
Riemann Sum = lim
||P|| 0
i=1
f(xi)Δxi
Where ||P|| is the width of the widest partition.
2
Example: Write
∫0 e
x
dx as a Riemann sum
i=n
Riemann Sum =
lim
||P|| 0
∑
i=1
exi Δxi
Group Practice
Look at the example problems on pages 247 through
250. Make sure the examples make sense. Work through
them with a friend.
Then look at the homework tonight and see if there are
any problems you think will be hard. Now is the time to
ask a friend or the teacher for help!
pp 251: 3, 4, 5, 6, 9, 11
Exit Question
What will the result be if we integrate sin(x) or cos(x)
from –x to x?
a) 0
b) 1
c) π
d) π/2
e) Not enough information
f) None of the above