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Calculus > Integrals > Flashcards

Integrals Flashcards

1
Q

Indefinite integral (∫____ dx)

A

Find the antiderivative of the ‘integrand’ contained

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2
Q

Integrand

A

The argument that is to be turned into the integral

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3
Q

Indefinite integrals to of a power

A

∫x^p dx=[x^(p+1)]/[p+1]+C

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

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4
Q

Integrals for functions multiplied by a constant

A

∫ c*f(x) dx=c(∫f(x) dx)

Same as that constant multiplied by the integral of that function

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5
Q

Integral of sums

A

∫ f(x)+g(x) dx=(∫f(x)dx)+(∫g(x)dx)

The integral is the sum of the integrals of each term

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6
Q

Integral of cos(ax)

A

∫cos(ax)dx= [sin(ax)]/a+C

Switched to sin and divided by ‘a’
Plus a constant

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7
Q

Integral of sin(ax)

A

∫cos(ax)dx= -[cos(ax)]/a+C

Switched to cos and divided by ‘-a’
Plus a constant

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8
Q

Integral of sec^2(ax)

A

∫sec^2(ax)dx= [tan(ax)]/a+C

Becomes tan(ax), and divided by a
Plus the constant
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9
Q

Integral of csc^2(ax)

A

∫csc^2(ax)dx= -[cot(ax)]/a+C

Becomes cot(ax), and divided by '-a'
Plus the constant
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10
Q

Integral e^(ax)

A

∫e^(ax)dx= [e^(ax)]/a+C

Same thing just divided by ‘a’
Plus a constant

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11
Q

Integral 1/x

A

∫(1/x)dx=ln|x|+C

Natural log (x) 
Plus a constant
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12
Q

Family of functions

A

Same integral but the constant varies

Go up and down on the y-axis

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13
Q

Integrals of acceleration

A

∫a(t)=v(t) Velocity

∫∫a(t)=s(t) Position

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14
Q

Integal of velocity

A

∫v(t)=s(t)

Position

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15
Q

Regular partition

A

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

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16
Q

Reimann sum

A

Sum of the areas of all regular partition rectangles
Σ(f(x)Δx)=(f(x1)Δx)+…+…
On the interval [a,b]

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17
Q

Left Reimann-Sum

A

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)

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18
Q

Right Reimann Sum

A

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)

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19
Q

Midpoint Reimann Sum

A

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)

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20
Q

Definite integral

A

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)

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21
Q

Closed integrals

A

If there is no interval, there is no integral

∫[a,a] f(x)dx=0

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22
Q

Reverse integrals

A

When the integral is flipped, it becomes negative

∫[a,b] f(x)dx= -∫[b,a] f(x)dx=0

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23
Q

Trapezoid area (under a linear function)

A

On the interval [a,b]

A(x)=Δx*[f(a)+f(b)]/2

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24
Q

Fundamental theorem of Calculus (Part 1)

A

Memorize word for word
“If f is continuous on [a,b], then the area function A(x)=∫[a,x] f(t)dt.
For a<b></b>

</b>

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25
Q

Fundamental theorem of Calculus (Part 2)

A

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)”

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26
Q

Integral of an odd function

A

If on the interval [-a,a]

Then always equal to 0

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27
Q

Integral of an even function

A

If on the interval [-a,a],

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

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28
Q

Average function value from integrals

A

(∫[a,b]f(x)dx)/(b-a)

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29
Q

Substituting within integrals

A 
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)]
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30
Q

Distance Traveled

A

s(b)-s(a)= |∫v(t)| = |∫∫a(t)|

On interval [a,b]

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31
Q

Position from velocity

A

s(t)=s(ti)+∫v(x)dx

interval [ti,t]

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32
Q

Velocity from Acceleration

A

v(t)=v(ti)+∫a(x)dx

interval [ti,t]

33
Q

‘Net’ values

A

f(b)-f(a)= ∫f’(t)dt

On interval [a,b]

34
Q

Future values of a function

A

f(t)=f(ti)+∫f’(x)dx

interval [ti,t]

35
Q

Region between two curves (from x)

A

A= ∫[f(x)-g(x)]dx

interval [a,b]

36
Q

Region between two curves (from y)

A

A= ∫[f(y)-g(y)]dy
interval [c,d]
When x=f(y) and x=g(y)
Needs practice

37
Q

Volume by slicing

A

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
Q

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

A

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
Q

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

A

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
Q

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

A

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
Q

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

A

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
Q

Arc length on an interval [a,b]

A

Arc= ∫√(1+f’(x)^2)dx

43
Q

Hooke’s Law (Spring Force)

A

F(x)=kx
K- spring coefficient, measure of the stiffness of that spring
X- distance pulled

44
Q

Work required to lift inside a container

A

W= ∫DGA(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 πr^2)
H(y)- Distance the water is lifted as a function of height ‘y’, represents net liquid height

45
Q

Pressure

A

P(y)=(DA(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
Q

Force of materials

A

F=∫DG(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
Q

Integral ∫1/x dx

A

Ln x

Simple as that

48
Q

Integral ∫ln x^p

A

∫ln x^p= p*(∫ln x)

49
Q

Integral ∫ln(x/y)

A

∫ln(x/y)= ∫ln(x)-∫ln(y)

50
Q

Integral ∫ln(xy)

A

∫ln(xy)= ∫ln(x)+∫ln(y)

51
Q

Integral ∫[f’(x)/f(x)]

A

∫[f’(x)/f(x)]= ln |f(x)|

52
Q

Integral ∫e^x

A

∫e^x= e^x

53
Q

Integral ∫b^x dx

A

∫b^x dx =(b^x)/(ln b) +C

54
Q

Integration by parts

A

∫f(x)g’(x)dx= f(x)*g(x)-∫g(x)f’(x)dx

55
Q

Integral ∫ln(x)

A

∫ln(x)dx=x*ln(x)-x+C

56
Q

Integral ∫sin^n(x)dx

A

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

Write it out

57
Q

Integral ∫cos^n(x)dx

A

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

Write it out

58
Q

Integral ∫tan^n(x)dx

A

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

59
Q

∫sec^n(x)dx

A

∫sec^n(x)dx =[sec^(n-2)xtan(x)]/(n-1)+(n-2)/(n-1)∫sec^(n-2)x dx
Write it out

60
Q

Integral ∫tan(x)dx

A

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

61
Q

Integral ∫cot(x)dx

A

∫cot(x)dx =ln|sin(x)|+C

62
Q

Integral ∫sec(x)dx

A

∫sec(x)dx =ln|sec(x)+tan(x)|+C

63
Q

Integral ∫csc(x)dx

A

∫csc(x)dx =-ln|csc(x)+cot(x)|+C

64
Q

Partial fraction decomposition (before integrating rational functions)

A

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
Q

Absolute Error

A

(ActualIntegral-RiemannSum)

66
Q

Relative error

A

(ActualIntegral-RiemannSum)/ActualIntegral

67
Q

Trapezoid Rule

A

Σ ([f(x-1)-f(x)]/2)Δx

68
Q

Integral Symmetry

A

D

69
Q

Integrals with Buoyance

A

D

70
Q

Archimede’s Principal

A

D

71
Q

Integrals Hyperbolic functions

A

D

72
Q

Euler’s method in integral

A

D

73
Q

Simpson’s Rule

A

F

74
Q

Preditor-prey integration models

A

G

75
Q

Mercator Projections

A

D

76
Q

Integrals in logistic growth

A

D

77
Q

Work

A

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)

78
Q

Antiderivatives F(x)

A

Basically the integral…
The antiderivative is the function to that derivative-function,
F’(x)=f(x)
Position is the antiderivative to a velocity curve

Calculus (17 decks)

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