Accelerated Calculus
Name: ___________________________
Due: 12/5/11
Properties of Definite Integral
“Rules” for Definite Integrals
Textbooks always give convenient boxes filled with rules and properties of definite integrals
Shame on them! These rules can be understood – and proved – by the most elementary
geometric thinking.
On a separate piece of paper, draw a picture that illustrates each of the following rules & exercises…
i. Order of Integration
ii. Zero
iii. Constant Multiple
iv. Sum and Difference
v. Additivity
b
a
f (x) dx f (x) dx
a
b
a
a
f (x) dx 0
(by definition)
k f (x) dx k f (x) dx
b
b
a
a
f (x) g(x) dx
b
b
a
a
b
a
f (x) dx
c
b
vi. Max-Min Inequality
f min (b a)
f (x) dx
f (x) dx
b
a
c
a
b
a
g(x) dx
f (x) dx
f (x) dx f max (b a)
if f (x) g(x) on [a, b] then
vii. Domination
(by definition)
b
a
f (x) dx
b
a
g(x) dx
viii. Average (Mean) Value
avg( f )
1
(b a)
b
a
f (x) dx
(only if f integrable on [a, b])
ix. Rule viii gets formalized a bit and turned into a theorem. Geometrically, what does it say?
The Mean Value Theorem (for Definite Integrals)
If f is continuous on [a, b], then at some point c in [a, b],
Ex1:
f (c)
1
(b a)
b
a
f (x) dx
Suppose we have a circle of radius r centered at the origin. We want to
know the average length of the chords perpendicular
to the diameter [-r, r]
on the x-axis.
a. Show that the length of the chord at x is 2 r 2 x 2 .
b. Set up an integral expression for the average value of 2 r 2 x 2 over the
interval [-r, r].
c. So, evaluate your integral to get the average length of a chord on a circle of
radius r.
Integrals apply to this problem?
d. How does the Mean Value Theorem for Definite
Ex2:
Consider the following scenario:
Questions: (let f (t ) represent the rate at which CO2 enters the water at time t)
t 6
a. Given this context, briefly describe in words what
f t dt tells us.
t 0
b. At what time was the CO2 level lowest? highest? Briefly justify your answers.
c. Estimate how much CO2 enters the pond during the night (from t 12 to t 24 ).
d. Estimate the CO2 level at dusk (twelve hours after dawn).
e. Does the CO2 level appear to be approximately in equilibrium? Briefly justify.
Ex 3:
Ex 4:
0
