L ectur e1 0| 1 Chapter 3
Integration
Outline 1. Antiderivatives and indefinite integral
2. Definite integrals
3. Riemann sum and areas 4. Fundamental theorems of
Calculus
L ectur e1 0| 2 1. Antiderivatives
Given a function ,
Q: What is a function having the rate of change equal to at any ?
Equivalently,
Q: What is such that
Examples
What is the displacement of a moving object having velocity :
What is the amount of charge in a circuit having current :
L ectur e1 0| 3 Def For a function , if satisfies
then is called an antiderivative of .
Theorem If is an antiderivative of on an interval , then any antiderivative of has the form
where is a constant.
The formula , which is the most general antiderivative of , is called the indefinite integral of denoted by
L ectur e1 0| 4 EX We have
Next,
Finally,
so
L ectur e1 0| 5 EX (Important antiderivative formulas) Show the following formulas
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Remark To find the indefinite integral of a given function , we find one function satisfying . Then
L ectur e1 0| 7 Rule Assume and and is a constant. Then
1. (Sum) is ATD of .
2. (Subtraction) is ATD of . 3. (Constant mult.) is ATD of .
Proof The statements are true because
L ectur e1 0| 8 EX (Employing formulas and rules)
Find the most general antiderivative for each of the following functions
L ectur e1 0| 9 EX (Applications) Find if
and .
L ectur e1 0| 10 2. Definite Integrals
The area problem
Let be a function and . Find the area of the region under the curve
L ectur e1 0| 11 EX (Approximate the are) Estimate the area under the curve from to .
Lower Riemann sum , Upper Riemann sum :
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From the table, one can expect that
Since for all , the area must be
We will justify this answer next.
L ectur e1 0| 14 Riemann Sum Let be a real-valued
function that is defined on .
Step 1. Divide equally into sub- intervals, that is
and
L ectur e1 0| 15 Step 2. Choose a representative point on each , that is
Riemann sum is defined to be
The limit
if exists, is called the (definite) integral of from to . It is denoted by
L ectur e1 0| 16 Area
If is a function on and on , the area under the curve from to is defined to be
provided the integral exists.
Remark Riemann sums and integrals are defined for any real-valued functions.
This means the function can have either positive or negative values.
L ectur e1 0| 17 Theorem Assume that is a piecewise continuous function, i.e. is continuous on or one can split into finitely many sub-intervals so that is continuous on each sub-interval.
Then the integral
exists.
L ectur e1 0| 18 EX Compute using the Riemann sum with .
Sol Divide into
Riemann sum is
So the integral is
L ectur e1 0| 19 Rules Assume and exist and is a constant. Then
L ectur e1 0| 20 EX Let
Evaluate
L ectur e1 0| 21 Rule Assume is a piecewise continuous function. Let . Then
L ectur e1 0| 22 EX Evaluate the integral if
has the graph as shown below.
