5.3/5.4 Definite Integrals
Tues Feb 23
1)
2)
Do Now
ò
Evaluate the following x
2 -
2 x dx x
-
1 x
2 dx

HW Review worksheet p.349
#3-6 35-38
• 3) 1.328125
• 4) 4.625
• 5) 2.7265625
• 6) 17.75
• 35) L = 1.81
• 36) L = .87
• 37) L = 3.8
• 38) L = 3.08
R = 1.67
R = .63
R = 4.32
R = 3.08

More and More Rectangles
• When approximating area, the more rectangles we use, the space is covered under the curve
• More rectangles = better approximation

Area Under a Curve
• For a function f defined on the interval
[a,b], if f is continuous on [a,b] and f(x) > 0 on [a,b], the area A under the curve y = f(x) on [a,b] is given by

Definite Integral
• The definite integral of f on the interval
[a,b] is for any function on the interval [a, b]

Fundamental Theorem of
Calculus
• If f is continuous on [a,b] and F(x) is the antiderivative of f, then
a b f ( x ) dx
=
F ( b )
-
F ( a )
• Step 1: Integrate f(x)
• Step 2: Plug in b and a into x
• Step 3: Subtract the 2 values

Notation
• Note:
F ( x ) b =
F ( b )
-
F ( a ) a
• The left side is often used instead of the right side

Area above and below x-axis
• Any area can be bounded by the x-axis and the function:
– If the area is above the x-axis, then it is considered positive
– If the area is below the x-axis, then it is considered negative

Reversing the Limits of
Integration
• If we reverse the limits of integration,
a b f ( x ) dx
= -
b a f ( x ) dx

Additivity for Adjacent Intervals
• Let a
£ b
£ c is integrable.
, and assume that f(x)
ò a c
Then: f ( x ) dx
ò a b f ( x ) dx
ò b c f ( x ) dx
This is useful for absolute value or piecewise functions

Examples
-
2
2
(10 x
9 +
3 x
5
) dx
3
5 e
2 x dx

You try
• 1)
• 2)
• 3)
• 4)
• 5)
ò
ò
1
4
0
2
( x
2 -
2 x ) dx
( x
-
1 x
2
) dx
ò p sin x dx
0
ò
0
4 e
-
2 x dx
ò
-
3
-
1
2 x dx

Closure
• Hand in: Compute the integral
ò
0
2
(2 x
-
3) dx
• HW: p.314 #5-37 odds

5.3 Definite Integral Practice
Wed Feb 24
• Do Now
• Evaluate each integral
• 1)
ò
0
8
( 3 x
x
2 / 3
) dx
• 2)
ò
0 p
/ 2
2sin x dx

• 5) 27/2
• 7) -1
• 9) 128
• 11) 27/2
• 13) 16/3
• 15) 31/40
• 17) 2/3
• 19) 12
• 21) 11/6
HW: p.314 #5-37 odds
23) 60 3
-
8
3
25) 2
27) 3/2
4
29)
31) 1
5
3 3
( 2
-
1)
33) e - 1
35)
1
6
( e
e
-
17
)
37) ln 5

Remember!
• When using trig functions and your calculator, you must use radians!

Practice
• Worksheet p.372 #6-11 14-22 27

Closure
• Hand in: Evaluate the integral
ò
-
2
0
(2 x
+ e x
) dx
• HW: Worksheet p.372 #6-11, 14-22 27

5.4 Fundamental Thm of
Calculus Pt 2
Thurs Feb 25
• Do Now
• Evaluate the integral
3
3
( x
sin x ) dx
0

HW Review p.372 #6-11
• 6) 3
• 7) 0
• 8) 8
• 9) 88/3
• 10) 5
• 11) 12/5

HW Review: p.372 #14-22, 27
• 14) 0
• 15) 2
-
1
• 16) 1
• 17) 3
• 18) 0
• 19)
• 20) e
+
1 e
-
2
2( e
e
-
1
)
21)
22)
3 e
6
2
6
+
1
3
-
( e
21
2
-
6
27) 16/3
-
1)

Fundamental Thm Part 2
• If f is continuous on [a,b] and then F
¢
( x )
= f ( x ) on [a,b]
F ( x )
=
ò a x f ( t ) dt

In words,
• If we differentiate an integral, they will cancel out.
• If the upper bound is a variable x, we can replace the current variable with x.
• NOTE: This is extremely helpful when we don ’ t know how to integrate something!

Fundamental Thm Part 2
• If the upper bound is something more than x, then we also need to multiply our answer by the derivative

Examples
• Ex 5.7 and 5.8
F (
• Find F ’ (x) x )
=
ò
1 x
( t
2 -
2 t
+
3) dt
2
F ( x )
=
2 x cos t dt

• 1)
• 2)
• 3)
You try: Find F ’(x)
F ( x )
=
ò
4 x
(2 t
3 t
+
2) dt
ò 2
F ( x )
=
1 x
( t
2 -
2 t
+
3) dt
2
F ( x )
=
1 x
(cos t
2
) dt

Closure
• Hand in: Find F ’ (x) if
F ( x )
=
1 x
( t
3 -
5 t
+
3) dt
• HW: p.320 #7-15 odds 21 23 29 31 33
• Quiz Tues March 1

5.4 Review/Practice
Fri Feb 26
• Do Now
• Calculate the derivatives
• 1) d dx
( t
2 + cos t ) dt
5 x
• 2) d dx
1 x
2 sin
2 t dt

HW Review: p.320 #7-15 21
• 7) 1
5 x
5 -
32
5
• 9) 1 – cos x
• 11)
• 13)
• 15)
• 21)
1
3
2
1 e
3 x
4
-
1
3 x
-
1
2 e
12
e
-
9 x
-
2 x
5
+ e
-
9 x
3
-
3 x
23 29-33
29)
31)
2 x
3 x
2 +
1
cos
4 s sin s
33)
2 x tan( x
2
)
-
1
2 x
-
1/2 tan x
1/2
• 23) sec(5t – 9)

5.1-5.4 Quiz Review
• 5.1 Approximation of Area under Curve
– Left, Right, Midpoint Approximations
• 5.3/4 Fundamental Theorem of Calculus
– Definite Integrals
– Derivatives of Integrals

Practice
• 5.3 (all) and 5.4 (skip 3,5) Worksheet

Closure
• What are the 2 parts of the
Fundamental Theorem of Calculus?
What can they be used for?
• Finish 5.3 5.4 worksheet
• 5.1-5.4 Tues March 1

5.1-5.4 Review
Mon Feb 29
• Do Now
• Evaluate each integral
• 1)
0
3 x
2 -
2 x
+
1 dx
• 2) Find the area under f(x) = x^2 on the interval [0, 2] using 4 right endpoint rectangles

HW Review: worksheet
• 5.3
• 1) 2
5.4
1a) 0
• 2a) 0
• 2b) 16/3
1b) tan(pi/4) = 1
2a)
• 2c) 20/9 + ln 9/4 2b)
1
3 x
3 -
1 x
2 sin x
2
-
2
2
-
2
3
• 2d) sqrt 2
• 2e) 28/3
4a)
4b) sin
3 x
2 x
2 tan x
3

5.1-5.4 Quiz Review
• Approximation of Area under Curve
– Left, Right, Midpoint Approximations
• Fundamental Theorem of Calculus
– Definite Integrals
– Derivatives of Integrals
• All rules from 4.9 (Antiderivatives) will be needed!

Last call for questions
• 4.9 – Antiderivatives
• 5.1 – Rectangle Approximations
• 5.2/5.3 Definite Integrals
• 5.4 Derivatives of Integrals

Closure
• Make up a possible quiz problem that you are confident you can solve…and solve it.
• Quiz Tomorrow