sec 5.4: indefinite integrals and the net change...
TRANSCRIPT
Sec 5.4: Indefinite Integrals and the Net Change TheoremWe 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 Looks like Result is It Means
DEFINITE
INDEFINITE
Example: Verify by differentiation that the following formula is correct.
∫cos3x dx = sin x - sin3x + C3
EXAMPLE: Evaluate each of the following. Pay attention to whether they are definite or indefinite integrals!
1. ∫sec x (sec x + tan x) dx
2. ∫(5ex - ∛4x) dx
3. ∫ dx3x - 2√x1
3
4. ∫ dθ1 - cos2θ2 sin θ
5. ∫ dxe2x + ex
0
1
ex
6. ∫ dy(2y - 5)2
1. ∫sec x (sec x + tan x) dx
2. ∫(5ex - ∛4x) dx
3. ∫ dx3x - 2√x1
3
4. ∫ dθ1 - cos2θ2 sin θ
5. ∫ dxe2x + ex
0
1
ex
6. ∫ dy(2y - 5)2
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 ?∫ f(x) dxab
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:
∫ F'(x) dx = F(b) - F(a)ab
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.
∫5n'(t) dt
0
1.
How many honeybees are there at the end of the fifth week?
= total bee population after 5 weeks
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.
2.
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.
3.
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?
∫ p'(t) dt10
20
