Transcript Limits and Derivatives
1 Chapter 5 – Integrals 5.2 The Definite Integral 5.2 The Definite Integral Georg Friedrich Bernhard Riemann 1826 - 1866
0 1 2 3 Review - Riemann Sum
V
1 8
t
2 1 When we find the area under a curve by adding rectangles, the answer is called a
Riemann sum
.
4 The width of a rectangle is called a
subinterval
.
1 2 3 subinterval The entire interval is called the
partition
.
partition Subintervals do not all have to be the same size.
5.2 The Definite Integral 2
3 Example 1 – pg. 382 # 4 sin
x
0 3 2 a) b) If you are given the information above, evaluate the Riemann sum with
n
=6, taking sample points to be right endpoints. What does the Riemann sum illustrate? Illustrate with a diagram.
Repeat part a with midpoints as sample points.
5.2 The Definite Integral
Idea of the Definite Integral lim
n
i n
1
f
i
x
is called the
definite integral
f
, of 4 If we use subintervals of equal length, then the length of a subinterval is:
b a a i x n x i
The definite integral is then given by:
n
lim
i n
1
i
x
5.2 The Definite Integral
5 Definite Integral in Leibnitz Notation
n
lim
i n
1
i
x
Leibnitz introduced a simpler notation for the definite integral:
n
lim
i n
1
i
a b
Note that the very small change in
x
becomes
dx
.
5.2 The Definite Integral
Explanation of the Notation upper limit of integration Integration Symbol
a b
integrand 6 variable of integration (dummy variable) lower limit of integration It is called a dummy variable because the answer does not depend on the variable chosen.
5.2 The Definite Integral
7 Theorem (3) If
f
is continuous on [
a
,
b
], or if
f
has only a finite number of jump discontinuities, then
f
is integrable on [
a
,
b
]; that is, the definite integral
a b
exists. 5.2 The Definite Integral
8 Theorem (4) Putting all of the ideas together, if
f
is differentiable on [
a, b
], then
n
lim
i n
1
i
a b
where
x i n
5.2 The Definite Integral
9 Example 2 Use the midpoint rule with the given value of
n
to approximate the integral. Round your answers to four decimal places.
0 /2 cos 4
xdx n
4 5.2 The Definite Integral
10 Evaluating Integrals using Sums 1.
i n
1
i
2.
i n
1
i
2 2 1) 1)(2
n
1) 6 3.
i n
1
i
3 2 1) 2 4.
i n
1
c
nc
5.
i n
1
ca i
c i n
1
a i
6.
i n
1
a i
b i
i n
1
a i
i n
1
b i
7.
i n
1
a i
b i
i n
1
a i
i n
1
b i
5.2 The Definite Integral
11 Example 4 – Page 377 #23 Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.
2 0
x
2 5.2 The Definite Integral
12 Example 5 Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.
1 2 3
x dx
5.2 The Definite Integral
13 Example 6 – Page 383 # 29 Express the integral as a limit of Riemann sums. Do not evaluate the limit.
2 6 1
x x
5
dx
5.2 The Definite Integral
14 Example 7 – page 383 # 17 Express the limit as a definite integral on the given interval.
n
lim
i n
1
x i
x i
2
x
[2, 6] 5.2 The Definite Integral
15 Example 8 – page 385 # 71 Express the limit as a definite integral.
n
lim
i n
1
i
4
n
5
x
4 5.2 The Definite Integral
16 Properties of the Integral 1.
a b cdx
2.
a b
)
a b
a b
3.
a b
4.
a b
c
a b
a b
a b
5.2 The Definite Integral
Properties Continued 5.
a c
c b
a b
6.
7.
0 for
a
( ) for
a b
then,
a b b
then,
a b
17 8.
If
m
)
a b
M
for
a
(
b
then, ) 5.2 The Definite Integral 0
a b
18 Example 9 – page 384 # 62 Use Property 8 to estimate the value of the integral.
0 2
x
3 3
x
3
dx
5.2 The Definite Integral
19 Example 10 – page 384 # 37 Evaluate the integral by interpreting it in terms of areas.
3 0 1 9
x
2
dx
5.2 The Definite Integral
20 Example 11 – page 383 # 28 Work in groups to prove the following:
a b
2
x dx
b
3
a
3 3 5.2 The Definite Integral
What to expect next… We will be evaluating Leibnitz integrals using the idea of antiderivatives and the fundamental theorem of calculus.
21 5.2 The Definite Integral