Mathematics - Lecture and practicenbogya/mathmgk/lecture_int_handout.pdf · De nite integral De nition Geometrically R b a f(x)dx is the signed area of the region that lies between

MathematicsLecture and practice

Norbert Bogya

University of Szeged, Bolyai Institute

2017

Norbert Bogya (Bolyai Institute) Mathematics 2017 1 / 23

Integration


Motivating question

What if?

We know the velocity function of an object, but we want toknow the position function.

We want to recover a function from its known rate of change.

Definition

A function F is an antiderivative of f on an interval I ifF ′(x) = f (x) for all x ∈ I .

The process of recovering a function F (x) from its derivative f (x) iscalled antidifferentiation.


Antiderivatives

Exercise

Find an antiderivative for each of the following functions.

(a) f (x) = 2x

(b) g(x) = 5x

(c) h(t) = 4t2 − 3t7

(d) h(x) =√x3 + 2

5√x2

Theorem

If F is an antiderivative of f on an interval I , then the most generalantiderivative of f on I is

F (x) + C

where C is an arbitrary constant.


Antiderivatives

Function Antiderivativec c · x

xα (α 6= −1)xα+1

α + 11/x log |x | (ln |x |)sin x − cos xcos x sin xex ex


Particular antiderivative

The previous theorem provides that the most general antiderivative off on I is a family of functions whose graphs are vertical translates ofone another. We can select a particular antiderivative from thisfamily by assigning a specific value to C .

Exercise

Find an antiderivative of f (x) =x3 − 2x

xthat satisfies F (1) = 2.


Linear substitution

Theorem

If F (x) is an antiderivative of f (x), thenF (ax + b)

ais an

antiderivative of f (ax + b).

Example

f (x) =√

3x − 5, F (x) =23(3x − 5)3/2

3

Exercise

Find the general antiderivative of each of the following functions.

(a) f (x) =√

2x − 1(b) g(x) = 5

√4x − 3

(c) h(x) =6

(5− x)2


Indefinite integral

Definition

The set of all antiderivatives of f is the indefinite integral of f withrespect to x , denoted by ∫

f (x)dx .

The symbol∫

is an integral sign. The function f is the integrand ofthe integral, and x is the variable of integration.

Example

(a)∫

(x + 1)dx =x2

2+ x + C , C ∈ R

(b)∫ √

2x − 1dx =23(2x − 1)3/2

2+ C , C ∈ R


Definite integral - Motivating question

What is the area of the shadedregion R that lies above thex-axis, below the graph ofy = 1− x2 and between thevertical lines x = 0 and x = 1?


Approximating Area

The area of a region with a curved boundary can be approximated bysumming the areas of a collection of rectangles. Using morerectangles can increase the accuracy of the approximation.

A(a) ≈ 1 · 12

+ 34· 12

= 78

= 0.875 A(b) ≈ 0.78125Norbert Bogya (Bolyai Institute) Mathematics 2017 10 / 23

Approximating Area

Lower estimation

A ≈ 0.634765625

Upper estimation

A ≈ 0.697265625

The exact area is between the lower and the upper estimation.


Notation

“Integral of f from a to b”∫ b

af (x) dx

a: Lower limit of integrationb: Upper limit of integrationf (x): the function is the integranddx : x is the variable of integration


Definite integral

Definition

Geometrically∫ b

af (x)dx is the signed area of the region that lies

between the x-axis, the graph of y = f (x) and the vertical lines x = aand x = b. Evaluation of the integral means finding the exact area.

Natural question: can this definite integral evaluated for everyfunction f (x)? (No.)

Theorem

A continuous function is integrable. That is, if a function f iscontinuous on an interval [a, b], then its definite integral over [a, b]exists.


Exercise

x

y

11

f

Find

∫ 2

−2f (x)dx .

x

y

11

f

A1 = 4 and A2 = −1.5∫ 2

−2f (x)dx = 4− 1.5 = 2.5


Properties of definite integrals


Mean value theorem for definite integrals

Theorem

If f is continuous on [a, b], thenat some point c ∈ [a, b],

f (c) =1

b − a

∫ b

a

f (x)dx .

Definition

The number1

b − a

∫ b

a

f (x)dx is called the integral mean (or

average) of f on the interval [a, b].


Exercise

x

y

11

f

Find the integral mean over the interval [−2, 2].

f [−2,2] =

∫ 2

−2 f (x)dx

2− (−2)=

2.5

4= 0.625


Connection between definite integral and

differentiation (Fundamental theorem of calculus)

F (x) =∫ x

af (t)dt

a ≤ x ≤ b

Theorem

If f is continuous on [a, b] thenF (x) =

∫ x

af (t)dt is continuous

on [a, b] and differentiable on(a, b) and its derivative is f (x);

F ′(x) = f (x).


Connection between definite integral and

differentiation (Fundamental theorem of calculus)

Theorem

If f is continuous at every point of [a, b] and F is any antiderivativeof f on [a, b], then ∫ b

a

f (x)dx = F (b)− F (a).

This theorem is also known as formula of Newton-Leibniz.


Examples

∫ 2

−3(6− x − x2)dx =

[6x − x2

2− x3

3

]2−3

= (12− 2− 8

3)− (−18− 9

2+

27

3) =

125

6

∫ 4

1

(3

2

√x − 4

x2

)dx =

[x3/2 +

4

x

]41

=

(43/2 +

4

4

)︸︷︷︸

8+1=9

−(

13/2 +4

1

)︸︷︷︸

1+4=5

= 4


Total area

IMPORTANT

Definite integral is not the total area.

To find the area between the graph of y = f (x) and the x-axis overthe interval [a, b], do the following.

(1) Subdivide [a, b] at the zeros of f .

(2) Integrate f over each subinterval.

(3) Add the absolute values of the integrals.

Exercise

Find the total area of the region between the x-axis and the graph off (x) = x3 − x2 − 2x , −1 ≤ x ≤ 2.


Physics

Exercise

The velocity of a moving body is described by the function

v(t) = t2 − t(ms

).

(a) Find the indefinite integral of v(t).

(b) Determine the position function s(t), such that s(0) = 1.

(c) Calculate the average velocity over the interval [1, 4].

(d) Find the extremal value of s(t).

(e) Find the acceleration function of the body.


Physics

Solutions:

(a)∫v(t)dt = t3

3− t2

2+ C

(b) s(t) = t3

3− t2

2+ 1

(c) v [1,4] =∫ 41 v(t)dt

4−1 = s(4)−s(1)3

=

(43

3− 42

2

)−( 1

3− 1

2)3

(d) Position function s(t) has extremum if and only if v(t) = 0 andv(t) changes sign.

v(t) = 0⇐⇒ t = 1 or t = 0

t < 0 t = 0 0 < t < 1 t = 1 1 < t

v + 0 − 0 +

s ↗ maxs(0)=1 ↘ min

s(1)=5/6 ↗(e) a(t) = 2t − 1


Documents

Mathematics - Lecture and practicenbogya/mathmgk/lecture_int_handout.pdf · De nite integral De nition Geometrically R b a f(x)dx is the signed area of the region that lies between