Download MATH 1325 – BUSINESS CALCULUS Section 11.4/11.5 The

M ATH 1325 – B USINESS C ALCULUS
Section 11.4/11.5 The Fundamental Theorem of Calculus/Evaluating Definite
Integrals, Part I
Establishes a connection between differential calculus and integral calculus.
Differentiation and integration are inverse processes.
The FTC makes integration easy.
The Fundamental Theorem of Calculus: If f is continuous on [a, b], then
Z b
f (x)dx = F(b) − F(a) where
a
F is any antiderivative of f . (F 0 = f )
Z b
f (x)dx by subtracting F at the endpoints.
If we know the antiderivative F of f , we can evaluate
a
NOTE: BE CAREFUL! There is a big difference between a definite integral and an indefinite integral!!
A definite integral is a
. An indefinite integral is a
Z 3
Ex: Evaluate
3x2 − 4xdx
−1
Properties of Integrals: Assume f and g are continuous functions. Then:
Z a
f (x)dx = 0
1.
a
Z b
2.
f (x)dx = −
Z a
a
f (x)dx
b
Z b
c f (x)dx = c f (x) where c is constant
3.
a
Z b
4.
[ f (x) ± g(x)]dx =
a
Z b
f (x)dx +
a
f (x)dx ±
a
Z c
5.
Z b
Z b
g(x)dx
a
Z b
f (x)dx =
c
f (x)dx for a < c < b
a
Z 13
Z 9
f (x)dx = 11 and
Ex: If it is known that
1
Z 13
f (x)dx = 5, find
1
f (x)dx.
9
.
Math 1325
Section 11.4 (with 11.5 – Part I) Continued
Ex: Find the area under f (x) = x2 − 6x + 10 from x = −1 to x = 2.
Ex: Find
Z 25 √
4
Ex: Find
2
√
x−
dx
x
Z 4 3
2x − 4x2 + 5x − 2
1
x2
dx
2
Math 1325
Section 11.4 (with 11.5 – Part I) Continued
Net Change: The net change in a function f over an interval [a, b] is given by f (b) − f (a) =
Z b
f 0 (x)dx,
a
provided f 0 is continuous on [a, b].
Ex: A division of Ditton Industries manufactures a deluxe toaster oven. Management has determined that the
daily marginal cost function associated with producing these toaster oven is given by
C0 (x) = 0.0003x2 − 0.12x + 20 where C0 (x) is measured in dollars per unit and x denotes the number of units
produced. Management has also determined that the daily fixed cost incurred in the production is $800.
(a) Find the total cost incurred by Ditton in producing the first 300 units of these toaster ovens per day.
(b) What is the total cost incurred by Ditton in producing the 201st through 300th units per day?
b
1
f (x)dx
b−a a
Ex: In a report issued by the Economist Intelligence Unit in 2013, the number of commercial vehicle registrations in the United States (in thousands) by year from 2010 through 2015 is approximated by the function
N(t) = 1.3926t 3 − 9.2873t 2 + 74.719t + 228.3(0 ≤ t ≤ 5) where t is measured in years with t = 0 corresponding to 2010. If the projection holds up, what would the approximate average commercial vehicle registration
per year be in the period from 2010 through 2015? (Round your answer to the nearest thousand.)
Z
Average Value of a Continuous Function f over [a, b]:
3