Math 142 Lecture Notes for Section
Section 6.5 -
1
The Fundamental Theorem of Calculus
Properties of Definite Integrals: If f and g are continuous functions, then
(i)
Rb
(ii)
Ra
(iii)
Rb
(iv)
Rb
a
a
a
a
cdx =
f (x)dx =
f (x)dx =
cf (x)dx =
Rb
f (x) ± g(x) =
Rb
(vi) If a ≤ c ≤ b, a f (x)dx =
(v)
a
Example 6.5.1:
R3
R3
R6
Suppose that 1 xdx = 4, 1 x2 dx = 26/3, and 3 x2 dx = 63. Use this information to
compute the following:
3
Z
(3x2 − 4x)dx =
(a)
1
Z
(b)
1
6
7x2 dx =
Math 142 Lecture Notes for Section
2
Fundamental Theorem of Calculus - Part 2: Suppose f is continuous on [a, b].
Then it holds that
Z b
dF
f (x)dx = F (b) − F (a), where F is any antiderivative of f, that is
= f.
dx
a
Example 6.5.2:
Evaluate the following:
3
Z
(x2 + 3)dx =
(a)
−2
Z
4
(ex + 3x)dx =
(b)
1
Z
(c)
2
(3x − 5)3 dx =
0
Net Change Theorem: The integral of a rate of change is the net change of the
original function:
Z b df
dx = f (x)|ba = f (b) − f (a).
dx
a
Math 142 Lecture Notes for Section
Example 6.5.3:
A rabbit population starts with 250 and increases at a rate of
does
Z 10 db
250 +
dt
dt
0
3
db
dt
bees per week. What
represent?
Example 6.5.4:
A flood starts at 2000 square feet
√ of a parking lot at time t = 0 and the water drains
dr
from the lot at a rate of dt = 3 t per second. How much water is drained during the
second minute?
Example 6.5.5:
A ball is fired out of a cannon at an initial velocity of 15m/s, with a rate of acceleration
measured by e−t/10 . How do we compute the velocity of the ball after 6 seconds?
Math 142 Lecture Notes for Section
4
Fundamental Theorem of Calculus - Part 1: Suppose f is continuous on [a, b]. If
Rb
dg
= f (x) for a ≤ x ≤ b.
g(x) = a f (t)dt, then g is the antiderivative of f , that is dx
This can also be written as
Z x
d
f (t)dt = f (x).
dx a
Example 6.5.6:
Find the derivative of the following functions:
(a) g(x) =
Rx
(b) k(x) =
Rx
(c) h(x) =
R x2
1
0
3
t2 dt.
2
et−t dt.
et dt.
Question: How do you average finitely many values?
Math 142 Lecture Notes for Section
5
Question: How do you average infinitely many values?
Definition 6.5.7:
The average value of a continuous function f on the interval [a, b] is given by
Z b
1
f (x)dx.
b−a a
Example 6.5.8:
Find the average value of the following functions on the given interval.
(a) f (x) = 3 on [1, 4].
(b) f (x) = 8x3 + x on [2, 6].
Example 6.5.9:
Given the supply function p = 20(e−.25x − 1), find the average price over the interval
[20, 40].
Section 6.5 Suggested Homework: 1-51 (odd), 55-63(odd)