6.1: Antidifferentiation
6.2 and 6.3: The definite integral
MTH 241: Business and Social Sciences Calculus
F. Patricia Medina
Department of Mathematics. Oregon State University
Chapter 6 (Sections 6.1, 6.2 and 6.3)
6.1: Antidifferentiation
6.2 and 6.3: The definite integral
Antidifferentiation
Definition 1
Suppose that f (x) is a given function and F(x) is a function having
f (x) as its derivative:
F 0 (x) = f (x).
We call F(x) and antiderivative of f (x).
Recall the example we discussed during last class...
6.1: Antidifferentiation
6.2 and 6.3: The definite integral
Proposition
If F 0 (x) = 0 for all x in the interval I, there is a constant C such that
F(x) = C for all x in I.
Observe that from this proposition, if F1 and F2 are both
antiderivatives for f and F(x) = F1 (x) − F2 (x) then
0
0
F 0 (x) = (F1 (x)) − (F2 (x)) = 0 and F1 and F2 just differ by a
constant.
We write
Z
f (x)dx = F(x) + C
to express that functions of the form F(x) + C are antiderivatives of f .
Z
The symbol
is called an integral sign.
Z
f (x)dx is the indefinite integral of f and represents the family
of antiderivatives of f .
6.1: Antidifferentiation
Examples
Verify that:
1
2
3
1 r+1
x + C, r 6= −1.
r+1
R kx
1
e dx = ekx + C, k 6= 0.
k
R 1
dx = ln |x| + C x 6= 0.
x
R r
x dx =
6.2 and 6.3: The definite integral
6.1: Antidifferentiation
6.2 and 6.3: The definite integral
Some properties of indefinite integrals
1
R
[f (x) + g(x)] dx = f (x) dx + g(x) dx.
R
R
2
R
kf (x) dx = k f (x) dx, where k is a constant.
R
Example
Compute
Z 3
1 √
3x
− 2e + + x .
x4
x
6.1: Antidifferentiation
6.2 and 6.3: The definite integral
Example: Application to Economics
A soap manufacturer estimates that its marginal cost of producing
soap powder is C0 (x) = .2x + 1 hundred dollars per ton at a production
level of x tons per day. Fixed costs are $ 200 per day. Find the cost of
producing x tons of soap powder per day.
6.1: Antidifferentiation
6.2 and 6.3: The definite integral
The definite integral
Suppose that f is continuous on an interval [a, b] with an antiderivative
F. The definite integral of f from a to b is
Z b
f (x) dx = F(b) − F(a).
a
Note that a and b are called the limits of integration. Here a is
the lower limit and b is the upper limit.
F(b) − F(a) is abbreviated by the symbol F(x)|ba
Rb
a
f (x) dx is the signed area between the graph of y = f (x) and
the x axis.
Example
Evaluate
Z 4
√
3 x dx.
1
6.1: Antidifferentiation
6.2 and 6.3: The definite integral
The graph of y = f (x) is given below. Evaluate:
Z 4
f (x) dx.
−4
6
4
2
−6
−4
−2
2
−2
−4
−6
4
6
6.1: Antidifferentiation
6.2 and 6.3: The definite integral
6
4
2
−6
−4
−2
2
−2
−4
−6
4
6
6.1: Antidifferentiation
6.2 and 6.3: The definite integral
More examples
Evaluate and display graphically the following definite integrals:
Z 1
x dx.
1
Z0 e
2
2ex dx.
1
Z 2
3
−2
x2 dx.
6.1: Antidifferentiation
6.2 and 6.3: The definite integral
Example: Application
A ball is thrown upward from a height of 256 feet above the ground,
with the initial velocity of 96 feet per second. The velocity at time t is
given by v(t) = 96 − 32t feet per second.
1
Find s(t), the function given the height of the ball.
2
How high will the ball go?
3
What is the distance traveled by the ball from time 1 to 3 seconds
after it was thrown?