Example

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Example. We can also evaluate a definite integral by interpretation of definite integral . Ex. Find by interpretation of definite integral.

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Presentation Transcript
Example
• We can also evaluate a definite integral by interpretation of definite integral.
• Ex. Findbyinterpretation of definite integral.
• Sol. By the interpretation of definite integral, we know the definite integral is the area of the region under the curve from 0 to a. From the graph, we see the region is a quarter disk with radius a and centered origin. Therefore,
Example
• Ex. Byinterpretation of definite integral, find
• Sol. (1)

(2)

Properties of definite integral
• Theorem(linearity of integral) Suppose f and g are

integrable on [a,b] and are constants, then

is integrable on [a,b] and

• Theorem(product integrability) Suppose f and g are integrable on [a,b], then is integrable on [a,b].
Properties of definite integral
• Theorem(additivity with respect to intervals)
• Remark In the above property, c can be any number, not necessarily between a and b.
• When the upper limit is less than the lower limit in the definite integral, it is understood as
• Especially,
Comparison properties of integral
• 1. If for then
• 2. If for then
• 3. If for then
• 4.
Estimation of definite integral
• Ex. Use the comparison properties to estimate the definite

integral

• Sol. Denote Then when

Letting we get the only critical number

By the closed interval method, we find the range for f(x):

Mean value theorems for integrals
• Second mean value theorem for integrals Let

g is integrable and on [a,b]. Then

there exists a number such that

• Proof. Let Since

we have and

Hence

or By intermediate value theorem

Mean value theorems for integrals
• First mean value theorem for integrals Let

then there exists a number such that

• Remark. We call the mean value of f on [a,b].
Example
• Ex. Suppose and

Prove that such that

• Proof. By the first mean value theorem for integrals, there

exists such that Thus

By Rolle’s theorem, such that

Function defined by definite integrals with varying limit
• Suppose f is integrable on [a,b]. For any given

the definite integralis a number. Letting x vary

between a and b, the definite integral defines a function:

• Ex. Find a formula for the definite integral with varying limit
• Sol. By interpretation of definite integral, we have
Properties of definite integral with varying limit
• Theorem(continuity) If f is integrable on [a,b], then the

definite integral with varying limit

is continuous on [a,b].

The fundamental theorem of calculus (I)
• The Fundamental Theorem of Calculus, Part 1 If f is

continuous on [a,b], then the definite integral with varying

limit is differentiable on [a,b] and

• Proof

is between x and as and

Therefore,

Definite integral with varying limits
• The definite integral with varying lower limit is

Since we have

• The most general form for a definite integral with varying

limits is To investigate its properties,

we can write it into the sum of two definite integrals with

varying upper limit

Definite integral with varying limits
• By the chain rule, we have the formula
Example
• Ex. Find derivatives of the following functions
• Sol.

(2) Let by chain rule,

Example
• Ex. Find derivative
• Sol.
• Ex. Find if
• Sol.
Example
• Ex. Find the limit
• Sol. By L’Hospital’s Rule and equivalent substitution,
• Question:
Example
• Ex. Find the limit
• Sol.
Example
• Ex. Suppose b>0, f continuous and increasing on [0,b].

Prove the inequality

• Sol. Let

Then F(0)=0 and when

This implies F(t) is increasing, thus

Example
• Ex. Suppose f is continuous and positive on [a,b]. Let

Prove that there is a unique solution in (a,b) to F(x)=0.

• Sol.
Fundamental theorem of calculus (II)
• The Fundamental Theorem of Calculus, Part 2 If f is

continuous on [a,b] and F is any antiderivative of f, then

• Proof Let then g is an antiderivative of f.

So F(x)=g(x)+C. Therefore,

• Remark The formula is called Newton-Leibnitz formula

and often written in the form

Example
• Ex. Evaluate
• Sol.
• Ex. Find the area under the parabola from 0 to 1.
• Sol.
Example
• Ex. Evaluate
• Sol.
• Ex. Evaluate
• Sol.
Example
• Anything wrong in the following calculation?
Differentiation and integration are inverse
• The fundamental theorem of calculus is summarized into
• The first formula says, when differentiation sign meets integral sign, they cancel out.
• The second formula says, first differentiate F, and then integrate the result, we arrive back to F.
Homework 12
• Section 5.1: 21
• Section 5.2: 22, 37, 53, 59, 67
• Section 5.3: 18, 50, 54, 62