Sec 5.4: Indefinite Integrals and the Net Change Theorem
We have a nice, compact notation for definite integrals and even a
way to show work when evaluating them.
Sometimes we need a similarly convenient notation to indicate the
antiderivative of a function. Currently, we'd have to write something
like
If f(x) = x2, then what is F(x)?
That's where indefinite integrals come in.
DEFINITION/NOTATION
The indefinite integral of a function f is
∫f(x)dx = F(x) + C
Remember this means that F'(x) = f(x).
Because you will be working with both definite and indefinite
integrals from now on in calculus, it's critical to remember
the difference between them.
Kind of Integral
DEFINITE
INDEFINITE
Looks like Result is
It Means Example: Verify by differentiation that the following
formula is correct.
3
3
∫cos x dx = sin x - sin x + C
3
EXAMPLE: Evaluate each of the following. Pay attention
to whether they are definite or indefinite integrals!
1
e2x + ex
1.
∫sec x (sec x + tan x) dx
5.
∫0
2.
∫(5ex - ∛4x) dx
6.
∫ (2y - 5)2 dy
3.
∫1
4.
3
∫
3x - 2
√x
dx
1 - cos2θ
dθ
2 sin θ
ex
dx
1.
2.
∫sec x (sec x + tan x) dx
∫(5ex - ∛4x) dx
3.
4.
3
∫1
∫
3x - 2
√x
dx
1 - cos2θ
dθ
2 sin θ
1
e2x + ex
5.
∫0
6.
∫ (2y - 5)2 dy
ex
dx
Interpreting the Definite Integral
We have seen that a definite integral can be used to
represent area under a curve.
What are other interpretations of
b
∫a
f(x) dx ?
If(x) is a graph of velocity and f(x) ≥ 0,then the integral represents the
displacement (change in position) from time t = a to time t = b.
Consider the units.
If v(t) is in m/sec and t is in seconds, what integral will find the displacement from 4 to 6 seconds?
If v(t) = 4 ­ x2 is the velocity of a particle moving in a straight line measured in ft/sec, find:
a.
the displacement of the particle from t=0 seconds to t = 3 sec.
b.
the total distance the particle travels from 0 to 3 seconds.
The Net Change Theorem
The integral of a rate of change is the net change in
the function F:
b
∫a
F'(x) dx = F(b) - F(a)
1. A honeybee population starts with 100 bees and increases at the rate of n'(t) bees per week. What does the following integral represent? Use a complete sentence and units.
5
n'(t) dt
∫0
How many honeybees are there at the end of the fifth week?
= total bee population after 5 weeks
2. The linear density of a rod of length 5 m is given by ρ(x) = 4 + x kg/m where x is the distance in meters from one end of the rod. Find the total mass of the rod.
3. Water flows from the bottom of a storage tank at a rate of 200 ­ 2t liters per minute, where 0≤t≤50. find the amount of water that flows from the tank during the first 10 minutes.
EXAMPLE: Suppose that p'(t) represents the rate at which the
population of a county increases, in people per year, starting in 1990.
Then what does the following integral represent?
20
∫10
p'(t) dt