Flashcards in Integrals Deck (78):

0

## Antiderivatives F(x)

###
Basically the integral...

The antiderivative is the function to that derivative-function,

F'(x)=f(x)

Position is the antiderivative to a velocity curve

1

## Indefinite integral (∫____ dx)

### Find the antiderivative of the 'integrand' contained

2

## Integrand

### The argument that is to be turned into the integral

3

## Indefinite integrals to of a power

###
∫x^p dx=[x^(p+1)]/[p+1]+C

Quantity x raised to one an additional power over the entire exponent

Plus a constant

4

## Integrals for functions multiplied by a constant

###
∫ c*f(x) dx=c(∫f(x) dx)

Same as that constant multiplied by the integral of that function

5

## Integral of sums

###
∫ f(x)+g(x) dx=(∫f(x)dx)+(∫g(x)dx)

The integral is the sum of the integrals of each term

6

## Integral of cos(ax)

###
∫cos(ax)dx= [sin(ax)]/a+C

Switched to sin and divided by 'a'

Plus a constant

7

## Integral of sin(ax)

###
∫cos(ax)dx= -[cos(ax)]/a+C

Switched to cos and divided by '-a'

Plus a constant

8

## Integral of sec^2(ax)

###
∫sec^2(ax)dx= [tan(ax)]/a+C

Becomes tan(ax), and divided by a

Plus the constant

9

## Integral of csc^2(ax)

###
∫csc^2(ax)dx= -[cot(ax)]/a+C

Becomes cot(ax), and divided by '-a'

Plus the constant

10

## Integral e^(ax)

###
∫e^(ax)dx= [e^(ax)]/a+C

Same thing just divided by 'a'

Plus a constant

11

## Integral 1/x

###
∫(1/x)dx=ln|x|+C

Natural log (x)

Plus a constant

12

## Family of functions

###
Same integral but the constant varies

Go up and down on the y-axis

13

## Integrals of acceleration

###
∫a(t)=v(t) Velocity

∫∫a(t)=s(t) Position

14

## Integal of velocity

###
∫v(t)=s(t)

Position

15

## Regular partition

### Integration method whereby the domain is divided into equally spaced rectangles, the midpoint of each reaches up to the output on the curve

16

## Reimann sum

###
Sum of the areas of all regular partition rectangles

Σ(f(x)*Δx)=(f(x1)*Δx)+...+...

On the interval [a,b]

17

## Left Reimann-Sum

###
Sum rectangles from the upper-left corner

Underestimates on positive intervals

Overestimates on negative intervals

n-measurements start at f(0) and increase by units of Δx

Σ(f(n)*Δx)

18

## Right Reimann Sum

###
Sum rectangles from the upper-right corner

Overestimates on positive intervals

Underestimates on negative intervals

n-measurements start at f(Δx) and continue to increase by units of Δx

Σ(f(n)*Δx)

19

## Midpoint Reimann Sum

###
Sum rectangles from the middle of the interval

Neither overestimates or underestimates on either interval

n-measurements start at f(Δx*.5) and continue to increase by units of Δx

Σ(f(n)*Δx)

20

## Definite integral

###
When n-measurements start at f(Δx*.5) and continue to increase by units of Δx

Lim as Δx→0 Σ(f(n)*Δx)=∫f(x)

Or

Lim as n→∞ Σ(f(n)*Δx)=∫f(x)

21

## Closed integrals

###
If there is no interval, there is no integral

∫[a,a] f(x)dx=0

22

## Reverse integrals

###
When the integral is flipped, it becomes negative

∫[a,b] f(x)dx= -∫[b,a] f(x)dx=0

23

## Trapezoid area (under a linear function)

###
On the interval [a,b]

A(x)=Δx*[f(a)+f(b)]/2

24

## Fundamental theorem of Calculus (Part 1)

###
Memorize word for word

"If f is continuous on [a,b], then the area function A(x)=∫[a,x] f(t)dt.

For a** **

###
Memorize word for word

###
If on the interval [-a,a]

###
If on the interval [-a,a],

###
When you have f(g(x)) you must also put g'(x)dx

###
s(b)-s(a)= |∫v(t)| = |∫∫a(t)|

###
s(t)=s(ti)+∫v(x)dx

###
v(t)=v(ti)+∫a(x)dx

###
f(b)-f(a)= ∫f'(t)dt

###
f(t)=f(ti)+∫f'(x)dx

###
A= ∫[f(x)-g(x)]dx

###
A= ∫[f(y)-g(y)]dy

###
When b-a equals the object width (cross section on the x-axis)

###
When b-a equals the object net width (cross section on the x-axis)

###
When b-a equals net object height (cross section on the y-axis)

###
When b-a equals half the object net width (cross section on the y-axis)

###
When b-a equals half the object net height (cross section on the x-axis)

###
Arc= ∫√(1+f'(x)^2)dx

###
F(x)=kx

###
W= ∫D*G*A(y)*H(y) dy

###
P(y)=(D*A(y)*H(y)*G)/S(y)

###
F=∫D*G*(a-y)*w(y) dy

##
Integral ∫1/x dx

###
Ln x

##
Integration by parts

##
Integral ∫sin^n(x)dx

###
∫sin^n(x)dx =[-sin^(n-1)x*cos(x)]/n+(n-1)/n*∫sin^(n-2)x dx

###
∫cos^n(x)dx =[cos^(n-1)x*sin(x)]/n+(n-1)/n*∫cos^(n-2)x dx

###
∫sec^n(x)dx =[sec^(n-2)x*tan(x)]/(n-1)+(n-2)/(n-1)*∫sec^(n-2)x dx

###
1) Completely factor denominator

###
The use of energy to create a force

25

## Fundamental theorem of Calculus (Part 2)

###
Memorize word for word

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

∫[a,b] f(x)dx =F(b)-F(a)"

26

## Integral of an odd function

###
If on the interval [-a,a]

Then always equal to 0

27

## Integral of an even function

###
If on the interval [-a,a],

then 2*([0,a] f(x)dx)

28

## Average function value from integrals

### (∫[a,b]f(x)dx)/(b-a)

29

## Substituting within integrals

###
When you have f(g(x)) you must also put g'(x)dx

Substitute u=g(x)

du=g'(x)dx

f(u)du

The interval [a,b] becomes [g(a),g(b)]

30

## Distance Traveled

###
s(b)-s(a)= |∫v(t)| = |∫∫a(t)|

On interval [a,b]

31

## Position from velocity

###
s(t)=s(ti)+∫v(x)dx

interval [ti,t]

32

## Velocity from Acceleration

###
v(t)=v(ti)+∫a(x)dx

interval [ti,t]

33

## 'Net' values

###
f(b)-f(a)= ∫f'(t)dt

On interval [a,b]

34

## Future values of a function

###
f(t)=f(ti)+∫f'(x)dx

interval [ti,t]

35

## Region between two curves (from x)

###
A= ∫[f(x)-g(x)]dx

interval [a,b]

36

## Region between two curves (from y)

###
A= ∫[f(y)-g(y)]dy

interval [c,d]

When x=f(y) and x=g(y)

Needs practice

37

## Volume by slicing

###
When b-a equals the object width (cross section on the x-axis)

And A(x) is the 'area under the curve' on a z-plane at the x-value

V= ∫A(x)dx

interval [a,b]

38

## Total Volume by washer method (axis-rotation arround x)

###
When b-a equals the object net width (cross section on the x-axis)

f(x) describes the outer 'hight' from the cross-section cut (outer radius)

g(x) describes the inner 'hight' from the cross-section cut (inner radius

Where f(x)-g(x) is the net height at x

V= ∫π*(f(x)^2-g(x)^2)dx

interval [a,b]

39

## Total Volume by washer method (axis-rotation arround y)

###
When b-a equals net object height (cross section on the y-axis)

f(y) describes the outer 'width' from the cross-section cut (outer radius)

g(y) describes the inner 'width' from the cross-section cut (inner radius)

V= ∫π*(f(y)^2-g(y)^2)dy

interval [a,b]

40

## Total Volume by shell method (axis-rotation arround y)

###
When b-a equals half the object net width (cross section on the y-axis)

f(x) describes the 'height' of the top of the object from the x-axis (x=inner radius at that f(x)

g(x) describes the 'height' of the bottom of the object from the x-axis (x=outer radius at that g(x)

Where f(x)-g(x) is the actual object net height at x

V= ∫2πx*(f(x)-g(x))dx

interval [a,b]

41

## Total Volume by shell method (axis-rotation arround x)

###
When b-a equals half the object net height (cross section on the x-axis)

f(y) describes the rightmost radius from the x-axis

g(y) describes the leftmost radius from the x-axis

V= ∫2πy*(f(y)-g(y))dy

interval [a,b]

42

## Arc length on an interval [a,b]

###
Arc= ∫√(1+f'(x)^2)dx

43

## Hooke's Law (Spring Force)

###
F(x)=kx

K- spring coefficient, measure of the stiffness of that spring

X- distance pulled

44

## Work required to lift inside a container

###
W= ∫D*G*A(y)*H(y) dy

On the interval [a,b]

D- material desity

G- gravity contant: 9.8 on earth, 0 in space

A(y)- Cross-sectional area as a function of height 'y' (usually π*r^2)

H(y)- Distance the water is lifted as a function of height 'y', represents net liquid height

45

## Pressure

###
P(y)=(D*A(y)*H(y)*G)/S(y)

D- material desity

G- gravity contant: 9.8 on earth, 0 in space

A(y)- Cross-sectional area as a function of height 'y' (usually π*r^2)

S(y)- surface area as a function of height'y'

H(y)- Distance the water is lifted as a function of height 'y', represents net liquid height

46

## Force of materials

###
F=∫D*G*(a-y)*w(y) dy

On the interval [0,a]

D- material desity

G- gravity contant: 9.8 on earth, 0 in space

(a-y)- net depth

w(y)- surface width as a function of height 'y'

47

##
Integral ∫1/x dx

###
Ln x

Simple as that

48

## Integral ∫ln x^p

### ∫ln x^p= p*(∫ln x)

49

## Integral ∫ln(x/y)

### ∫ln(x/y)= ∫ln(x)-∫ln(y)

50

## Integral ∫ln(xy)

### ∫ln(xy)= ∫ln(x)+∫ln(y)

51

## Integral ∫[f'(x)/f(x)]

### ∫[f'(x)/f(x)]= ln |f(x)|

52

## Integral ∫e^x

### ∫e^x= e^x

53

## Integral ∫b^x dx

### ∫b^x dx =(b^x)/(ln b) +C

54

##
Integration by parts

### ∫f(x)g'(x)dx= f(x)*g(x)-∫g(x)f'(x)dx

55

## Integral ∫ln(x)

### ∫ln(x)dx=x*ln(x)-x+C

56

##
Integral ∫sin^n(x)dx

###
∫sin^n(x)dx =[-sin^(n-1)x*cos(x)]/n+(n-1)/n*∫sin^(n-2)x dx

Write it out

57

## Integral ∫cos^n(x)dx

###
∫cos^n(x)dx =[cos^(n-1)x*sin(x)]/n+(n-1)/n*∫cos^(n-2)x dx

Write it out

58

## Integral ∫tan^n(x)dx

### ∫tan^n(x)dx= [tan^(n-1)x]/(n-1)-∫tan^(n-2)x dx

59

## ∫sec^n(x)dx

###
∫sec^n(x)dx =[sec^(n-2)x*tan(x)]/(n-1)+(n-2)/(n-1)*∫sec^(n-2)x dx

Write it out

60

## Integral ∫tan(x)dx

### ∫tan(x)dx =-ln|cos(x)|+C =ln|sec(x)|+C

61

## Integral ∫cot(x)dx

### ∫cot(x)dx =ln|sin(x)|+C

62

## Integral ∫sec(x)dx

### ∫sec(x)dx =ln|sec(x)+tan(x)|+C

63

## Integral ∫csc(x)dx

### ∫csc(x)dx =-ln|csc(x)+cot(x)|+C

64

## Partial fraction decomposition (before integrating rational functions)

###
1) Completely factor denominator

2) Set original function equal to sum of denominator factors with alphabetical (A, B, C, ...) numerators

3) work the algebra

65

## Absolute Error

### (ActualIntegral-RiemannSum)

66

## Relative error

### (ActualIntegral-RiemannSum)/ActualIntegral

67

## Trapezoid Rule

### Σ ([f(x-1)-f(x)]/2)Δx

68

## Integral Symmetry

### D

69

## Integrals with Buoyance

### D

70

## Archimede's Principal

### D

71

## Integrals Hyperbolic functions

### D

72

## Euler's method in integral

### D

73

## Simpson's Rule

### F

74

## Preditor-prey integration models

### G

75

## Mercator Projections

### D

76

## Integrals in logistic growth

### D

77

## Work

###
The use of energy to create a force

Given in NewtonMeters, the product of force and displacement or the integral of Force as a function of displacement when graphed against eachother

W=F*Δs=∫F(Δs)

**
**