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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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example
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,
example1
Example
  • Ex. Byinterpretation of definite integral, find
  • Sol. (1)

(2)

properties of definite integral
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 integral1
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
Comparison properties of integral
  • 1. If for then
  • 2. If for then
  • 3. If for then
  • 4.
estimation of definite integral
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
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 integrals1
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].
example2
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
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
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 (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
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 limits1
Definite integral with varying limits
  • By the chain rule, we have the formula
example3
Example
  • Ex. Find derivatives of the following functions
  • Sol.

(2) Let by chain rule,

example4
Example
  • Ex. Find derivative
  • Sol.
  • Ex. Find if
  • Sol.
example5
Example
  • Ex. Find the limit
  • Sol. By L’Hospital’s Rule and equivalent substitution,
  • Question:
example6
Example
  • Ex. Find the limit
  • Sol.
example7
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

example8
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
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

example9
Example
  • Ex. Evaluate
  • Sol.
  • Ex. Find the area under the parabola from 0 to 1.
  • Sol.
example10
Example
  • Ex. Evaluate
  • Sol.
  • Ex. Evaluate
  • Sol.
example11
Example
  • Anything wrong in the following calculation?
differentiation and integration are inverse
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
Homework 12
  • Section 5.1: 21
  • Section 5.2: 22, 37, 53, 59, 67
  • Section 5.3: 18, 50, 54, 62