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### Basics of Integration

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Integration: First Principles

»  cause-effect relation in two quantities

→  eg: speed-displacement

»  The cause is calculated as a function of an algebraic expression in a variable.

→  eg: speed v=3t^2+2

»  The effect is derived to be "aggregate of change of cause with respect to the variable".

→  eg : displacement = aggregate of change of displacement

»  In such a case, the effect is another algebraic expression in the variable.

→  displacement is a function of t

»  The effect is computed as aggregate of change : the sum of change over an interval of the variable.

→  displacement =lim_(n->oo) sum_(i=1)^n v((i x)/n) x/n

»  Integration or integral of a function

In an interval from 0 to x, the aggregate of change of function f(x) is the integral of the function.

int f(x) dx

= c + int_0^x f(x)dx

= lim_(n->oo) sum_(i=1)^n f((i x)/n) x/n

### Integration: First Principles

plain and simple summary

nub

plain and simple summary

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The aggregate of change of a function with respect to the variable is the integration of the function.

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Starting on learning "First Principles of integration". In this page, integration is defined in first principles.

A car travels at speed 10m//s. What is the distance traveled in 2 seconds?

• 20m
• 10m

The answer is "20m". Speed multiplied by time gives the distance traveled in that time.

A car travels at speed 10m//s in the first 2 seconds, and at speed 15m//s in the next 3 seconds. What is the total distance traveled?

• distance =15 xx 3 = 45m
• distance = 10 xx 2 + 15 xx 3 = 65 m

The answer is "distance = 10 xx 2 + 15 xx 3 = 65 m"

Distance is computed as speed (cause) repeatedly added for the time durations (aggregate).

distance = speed1 xx time1 + speed2 xx time2 +...

Distance is the aggregate of speed. It is generalized as aggregate of change.

A car is moving at speed 20m/sec. What is the distance covered in t sec ?

• without a numerical value for t the distance traveled cannot be computed.
• For any value of t, the distance traveled is s=20t meter

The answer is "For any value of t, the distance traveled is s=20t meter".

Measurement can be expressed as a function of a variable.

A car is moving with speed given as a function of time v=3t^2+2 m/sec. What is the speed at t=2sec?

• 14 meter/sec
• 6 meter/sec

The answer is "14 meter/sec". Substitute t=2 in the formula for v.

A car is moving with speed given as a function of time v=3t^2+2 m/sec. Couple of students want to calculate the distance traveled in 2 sec. We know that

text(distance) = text(speed) xx text(time)

•  Person A does the following: At t=2s, the speed is 14m/sec. So the distance is 14 xx 2 = 28m.

•  Person B does the following: At t=1s, the speed is 5m/sec. And at t=2s, the speed is 14m/sec. So the distance traveled is 5xx1+14xx(2-1) = 19 m.

Which one is correct?

• Person A
• Person B
• Neither of them

The answer is "Neither of them". The solution is explained in the subsequent questions.

A car is moving with speed given as a function of time v=3t^2+2 m/sec. We know that

text(distance) = text(speed) xx text(time)

One student wants to calculate the distance traveled.

•  At t=0 second, the speed is =3xx0+2 = 2m/sec.

•  At t=1 second, the speed is = 3xx1+2 = 5m/sec.

•  At t=2 second, the speed is =3xx4+2 = 14m/sec.

Speed varies with time. So, consider time intervals 0 to 1sec and 1sec to 2sec. In each of these time intervals, the average of start and end speed can be taken as the speed during the time interval.

The speed during time 0<t<1 =(2+5)/2 = 3.5 m/sec

The speed during time 1<t<2 =(5+14)/2 = 9.5 m/sec

So distance traveled is =3.5xx1+9.5xx(2-1) = 13m

What is calculated in this?

• Approximate value of distance traveled
• accurate value of distance traveled

The answer is "Approximate value of distance traveled". It is approximate, because the speed continuously changes with time, but the calculation approximates to average over 1 second intervals.

A car is moving with speed given as a function of time v=3t^2+2. Which of the following is expected?

• displacement is a numerical value
• displacement is a function of time

The answer is "displacement is a function of time"

A car is moving with speed given as a function of time v=3t^2+2. Which of the following is correct?

• Displacement is aggregate of change of speed. So, one has to figure out how to find the aggregate of change of an algebraic expression.
• Displacement is defined only when speed during a time period is given. If the speed is given as a function of a variable, displacement cannot be calculated.

The answer is "Displacement is aggregate of change of speed. So, one has to figure out how to find the aggregate of change of an algebraic expression".

A car is moving with speed given as a function of time v=3t^2+2. Displacement is aggregate-of-change. Few students are set to find the aggregate of change
students may work these out to understand

•  Person A found aggregate of change, displacement, for 1 second interval. sum_(i=1,2,3,...)^(t) (3i^2+2)xx1

•  Person B found aggregate of change, displacement, for 2 second interval. sum_(i=2,4,6,...)^(t) (3i^2+2)xx2

•  Person C found aggregate of change, displacement, for 0.5 second interval. sum_(i=0.5,1,1.5,...)^(t) (3i^2+2)xx0.5

What is actually calculated in each?

• approximation of the distance traveled
• continuous-aggregate displacement

The answer is "approximation of the distance traveled"

A car is moving with speed given as a function of time v=3t^2+2. An odometer is attached to a wheel. The odometer measures the rotation of the wheel and proportionally provides the distance traveled by the car. What is the distance shown in the odometer?

• approximate distance traveled by the car
• continuous aggregate distance traveled for any given time

The answer is "continuous-aggregate distance traveled for any given time".

A car is moving with speed given as a function of time v(t) = 3t^2+2. An odometer can be used to measure the continuous aggregate distance traveled by the car.

The approximate distance can be computed with time interval delta is
sum_(i=delta, 2delta,3delta,...)^(t) (3i^2+2)xx delta
where delta=t/n,
n is the number of steps between 0 and t

When is the error in the approximation reduced?

• when the number of steps n is small
• when the number of steps n is large

The answer is "when the number of steps n is large". The speed changes with time. As the step-size is made smaller, the speed is approximated better. For smaller step size, the total number of steps has to be higher.

A car is moving with speed given as a function of time v(t) = 3t^2+2. An odometer can be used to measure the continuous-aggregate distance traveled by the car.

The approximate distance can be computed with time interval delta is
sum_(i=delta, 2delta,3delta,...)^(t) (3i^2+2)xx delta
where delta=t/n,
n is the number of steps between 0 and t

When is the algebraic expression exactly the continuous-aggregate distance?

• when the number of steps n is oo
• when the step width delta is 0
• both the above

The answer is "both the above". This is a big jump in understanding. The speed is continuously changing and the summation is done continuously (not in steps) when n is oo.

Generalizing that, for a function f(x) the approximate aggregate of change is sum_(i=delta,2delta,3delta,...)^(x) f(i)xx delta
where delta = x/n

The same can be given as sum_(i=1,2,3,...)^(n) f((i x)/n)xx x/n
where i takes positive integer values.

The same is simplified as sum_(i=1)^(n) f((i x)/n)xx x/n

A car is moving with speed given as a function of time v(t) = 3t^2+2. The approximate distance = sum_(i=1)^(n) (3((i t)/n)^2+2)xx t/n

When n=oo, the expression gives continuous-aggregate distance
delta=t/n = 0

text(distance) = sum_(i=1)^(oo) (3xx 0^2+2)xx 0
=0+0+0+... oo text( times)
=0 xx oo

What is the value of 0 xx oo?

• 0
• oo
• indeterminate value 0//0

The answer is "indeterminate value 0//0".

A car is moving with speed given as a function of time v(t) = 3t^2+2. The approximate distance = sum_(i=1)^(n) (3((i t)/n)^2+2)xx t/n

When n=oo, the expression gives continuous-aggregate distance as 0//0: indeterminate value.

How does one solve a function evaluating to indeterminate value?

• Use Limit of the function as n approaching oo.
• there is no method to solve a function evaluating to 0//0

The answer is "Use Limit of the function as n approaching oo."

A car is moving with speed given as a function of time v(t) = 3t^2+2. The approximate distance = sum_(i=1)^(n) (3((it)/n)^2+2)xx t/n
n is the number of steps between 0 and t

When n=oo, the expression gives continuous-aggregate distance
= sum_(i=1)^(n) (3((it)/n)^2+2)xx t/n |_(n=oo)
=sum_(i=1, 2...)^(n) 0
=0xxoo = 0/0

Since the distance evaluates to indeterminate value, the limit is used to check if the function is defined.

lim_(n->oo) sum_(i=1)^(n) (3((it)/n)^2+2)xx t/n

= lim_(n->oo) sum_(i=1)^(n) (3((it)/n)^2)xx t/n + lim_(n->oo) sum_(i=1)^(n) (2)xx t/n

= lim_(n->oo) (3t^3)/n^3 sum_(i=1)^(n) i^2 + lim_(n->oo) (2t)/n sum_(i=1)^(n) 1

substituting sum_(i=1)^(n) i^2 = n(2n+1)(n+1)//6
and sum_(i=1)^(n) 1 = n
=lim_(n->oo) (3t^3)/n^3 xx (n(2n+1)(n+1))/6 + lim_(n->oo) (2t)/n xx n

=lim_(n->oo) (3t^3)/n^3 xx (2n^3 + 3n^2 + n)/6 + lim_(n->oo)(2t)

=lim_(n->oo) (3t^3)/6 xx (2 + 3/n + 1/n^2) + lim_(n->oo) 2t

applying limit
=(3t^3)/6 xx (2 + 0 + 0) + 2t

=t^3 + 2t

The continuous-aggregate distance is computed as an algebraic expression.

A car is moving with speed given as a function of time v(t) = 3t^2+2. The approximate distance = sum_(i=1)^(n) (3((it)/n)^2+2)xx t/n
n is the number of steps between 0 and t

The continuous-aggregate distance traveled = lim_(n->oo) sum_(i=1)^(n) (3((it)/n)^2+2)xx t/n

Note that the distance traveled is computed starting from time t=0, at which point, the initial distance of the car can be non-zero. So,
the distance = initial distance at time 0 + distance traveled between time 0 and t.
For the given problem, the distance =c + lim_(n->oo) sum_(i=1)^(n) (3((it)/n)^2+2)xx t/n
where c is a constant.

Summarizing the learning so far,

•  Two quantities are in a cause-effect relation.

•  the cause is calculated as a function of an algebraic expression in a variable.

•  The effect is derived to be "aggregate of change of cause with respect to the variable".
(note: there are other forms of relation between cause-effect, such as multiple, addition, exponent. In this topic, we are concerned with only the aggregate of change relation.)

•  In such a case, the effect is another algebraic expression in the variable.

•  The effect is computed as, the continuous aggregate of change: the aggregate of the cause, over an interval, with the interval split into infinite partitions. This calculation is named as integration or integral of the function.

Note 1: The summation in integral has many other forms, Riemann, Lebesgue, and Darboux forms, which will be introduced in due course.

Note 2: Differentiation is instantaneous rate of change or rate-of-change. Integration is continuous aggregate of change or aggregate-of-change

Which of the following is a meaning for the word 'integrate'?

• combine one with another to form a whole
• convert a fraction to an integer

The answer is 'combine one with another to form a whole'. The aggregate of change is aptly named integral or integration.

What is the term used to refer "continuous-aggregate" of a function ?

Integration in the context of cause-effect pair in continuous aggregate relation:
If the cause is given by f(x) then the effect is computed as integration or integral of f(x) denoted as
int f(x) dx

= c + lim_(n->oo) sum_(i=1)^(n) f((ix)/n)xx x/n

The second term is understood to be continuous aggregate between 0 and x. This is denoted as
int f(x) dx

= c + int_0^x f(x)dx
= c + lim_(n->oo) sum_(i=1)^(n) f((ix)/n)xx x/n

Cause-effect was explained to understand the physical significance. Abstracting this and understanding the quantities involved in integration:

A quantity u=f(x) is related to another quantity v such that v is the aggregate of change of u with respect to x. Then,
v

=int u dx

=c + int_0^x u dx

= c + lim_(n->oo) sum_(i=1)^(n) f((ix)/n)xx x/n

Note that int u dx is another quantity v, related to the given quantity u.

• f(x) is called integrand.

• x is the variable of integration.

• c is the constant of integration.

• in int_0^x, the value 0 denotes the start position of integration called "lower limit" of integration.

• in int_0^x, the variable x denotes the end position of integration called "upper limit" of integration.

What is aggregate of change of a function called?

The answer is "integration or integral of the function".

Students can connect the notation int f(x) dx as

•  the small difference in x is given as dx

•  the multiplication by f(x) to dx implies, the value of function for the small difference is multiplied.

•  int denotes the sum with limit of dx becoming close to 0.

int f(x) dx = c + int_0^x f(x)dx

•  The left-hand-side gives the general form of integration. it is later explained as anti-derivative or indefinite integral.

•  The c, the constant of integration, is the initial value of the result.

•  The 0 and x are the starting point and terminal points of the summation.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Integral or Integration of a function : The aggregate of change of the function f(x) is defined as.

int f(x) dx

=c+ int_0^x f(x) dx

=c + lim_(n->oo) sum_(i=1)^(n) f((ix)/n)xx x/n

note1: The definite integrals and anti-derivatives or indefinite integrals are introduced in due course.
note2: The summation has many other forms, Riemann, Lebesgue, and Darboux forms, which will be introduced in due course.

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Finding the integral of y=7x in first principles:

int 7x dx

= c + lim_(n->oo) sum_(i=1)^(n) ((7ix)/n)xx x/n

= c + lim_(n->oo) sum_(i=1)^(n) ((7ix)/n)xx x/n

= c + lim_(n->oo) (7x^2)/n^2 sum_(i=1)^(n) i

substituting sum_(i=1)^(n) i = n(n+1)//2
= c + lim_(n->oo) (7x^2)/n^2 xx (n(n+1))/2

= c + lim_(n->oo) (7x^2)/n^2 xx (n^2 + n)/2

= c + lim_(n->oo) (7x^2)/2 xx (1 + 1/n )

applying limit
= c + (7x^2)/2 xx (1 + 0 )

= c + 7x^2//2

What does the above prove?

• int 7x dx = c + 7x^2 //2
• int 7x dx = c + 0+0

The answer is "int 7x dx = c + 7x^2 //2"

Finding the integral of y=sinx in first principles:

int sinx dx

lim_(n->oo) sum_(i=1)^(n) (sin((ix)/n))xx x/n

= lim_(n->oo) x/n sum_(i=1)^(n) sin((ix)/n)

Is it possible to simplify this and resolve the indeterminate value 0//0 of limit?

• the summation is not easily simplified
• it is not easily solved in the forward summation
• it can be easily solved based on some property of the limit of summation
• all the above

The answer is "all the above". Many functions are not easily solved by forward summation as defined by the first principles. We will learn properties of the limit of summation to solve integration of such functions.

Progress

Progress

A car travels at speed 10 meter per second. What is the distance traveled in 2 seconds.
20
20 meter
10
10 meter
The answer is "20 meter". Speed multiplied by time gives the distance traveled in that time.
A car travels at speed 10 meter per second in the first 2 seconds, and at speed 15 meter per second in the next 3 seconds. What is the total distance traveled.
15;3;45
distance = 15 multiplied 3 = 45 meter
10;2;65
distance = 10 multiplied 2 + 15 multiplied 3 = 65 meter
The answer is "distance = 65 meter"
Distance is computed as speed repeatedly added for the time durations. ;; Distance = speed 1 multiplied time 1, + speed 2 multiplied time 2 et cetra. ;; Distance is the aggregate of speed. It is generalized as aggregate of change.
A car is moving at speed 20 meter per second. What is the distance covered in t seconds.
without;numerical;t;tea;cannot;computed
without a numerical value for t the distance traveled cannot be computed.
any; 20;twenty
For any value of t , the distance traveled is s=20t meter
The answer is "For any value of t, the distance traveled is s=20 t meter". Measurement can be expressed as a function of a variable.
A car is moving with speed given as a function of time v = 3 t squared + 2 meter per second. What is the speed at t = 2 second.
fourteen;14
14 meter per second
six;6
6 meter per second
The answer is "14 meter per second".
A car is moving with speed given as a function fo time v = 3 t squared + 2 meter per second. Couple of students want to calculate the distance traveled in 2 second. We know that distance = speed multiplied time. ;; Person A does the following. At t = 2 seconds, the speed is 14 meter per second. So the distance is 14 multiplied 2 = 28 meter.;; Person B does the following: at t = 1 second, the speed is 5 meter per second. And at t=2 second, the speed is 14 meter per second. So the distance traveled is, 5 multiplied 1, + 14 multiplied 2 minus 1, = 19 meter. ;; which one is correct.
a;A
Person A
b;bee
Person B
neither;them
Neither of them
The answer is "Neither of them". The solution is explained in the subsequent questions.
A car is moving with speed given as function of time. ;; One student wants to calculate the distance traveled. The speed is calculated for time t=0, t=1, t=2 as 2, 5 and 14 respectively. Speed varies with time. So, consider time intervals 0 to 1 second and 1 to 2 second. In each of these time intervals, the average of start and end speed can be taken as the speed during the time interval. ;; In that, the distance traveled is calculated as 13 meter. What is calculated in this
approximate
Approximate value of distance traveled
accurate
accurate value of distance traveled
The answer is "Approximate value of distance traveled". It is approximate, because the speed continuously changes with time, but the calculation approximates to average over 1 second intervals.
A car is moving with speed given as a function of time. Which of the following is expected.
numerical;value
displacement is a numerical value
function;time
displacement is a function of time
The answer is "displacement is a function of time"
A car is moving with speed given as a function of time. Which of the following is correct.
aggregate;change;figure;out;how;to ;expression;algebraic
Displacement is aggregate of change of speed. So, one has to figure out how to find the aggregate of change of an algebraic expression.
distance;only;period;cannot
Displacement is defined only when speed during a time period is given. If the speed is given as a function of a variable, displacement cannot be calculated.
The answer is "Displacement is aggregate of change of speed. So, one has to figure out how to find the aggregate of change of an algebraic expression".
A car is moving with speed given as a function of time. Displacement is aggregate of change. Few students are set to find the aggregate of change. Person A found aggregate of change for 1 second interval. Person B found aggregate of change for 2 second intervals. Person C found aggregate of change for point 5 second intervals. What is actually calculated in each
approximation;distance;traveled
approximation of the distance traveled
continuous;aggregate;displacement
continuous-aggregate displacement
The answer is "approximation of the distance traveled"
A car is moving with speed given as a function of time. An odometer is attached to a wheel. The odometer measures the rotation of the wheel and proportionally provides the distance traveled by the car. What is the distance shown in the odometer.
approximate;distance;traveled
approximate distance traveled by the car
continuous;aggregate;given;any
continuous aggregate distance traveled for any given time
The answer is "continuous-aggregate distance traveled for any given time".
A car is moving with speed given as a function of time. An odometer can be used to measure the continuous aggregate distance traveled by the car. The approximate distance can be computed with time interval delta is summation over i, equal delta, 2 delta, et cetera, till t, 3 i squared + 2, multiplied delta. Where delta = t over n, n is the number of steps between 0 and t. When the error in the approximation is reduced?
small
when the number of steps n is small
large
when the number of steps n is large
The answer is "when the number of steps n is large". The speed changes with time. As the step-size is made smaller, the speed is approximated better. For smaller step size, the total number of steps has to be higher.
A car is moving with speed given as a function of time. An odometer can be used to measure the continuous-aggregate distance traveled by the car. The approximate distance can be computed with time interval delta. ;; when is the algebraic expression exactly the
infinity;n;en
when the number of steps n is infinity
step;width
when the step width delta is 0
both;above
both the above
The answer is "both the above". This is a big jump in understanding. The speed is continuously changing and the summation is done continuously when n is infinity.
Generalizing that, for a function f of x, the approximate aggregate of change is sum over, i equals 1 to n, f of i x by n, multiplied x by n. ;; The simplified expression of this is summation over i = i, to n, f of i x by n, multiplied x by n.
A car is moving with speed given as a function of time. The approximate distance is given by a sum. When n equals infinity, the expression gives continuous-aggregate distance. It is calculated as 0 multiplied infinity. What is the value of 0 multiplied infinity.
0
0
infinity
infinity
indeterminate;value
indeterminate value 0 by 0
The answer is "indeterminate value 0 by 0"
A car is moving with speed given as a function of time. Approximate distance is given by a sum. When n equals infinity, the expression gives continuous-aggregate distance as 0 by 0 : indeterminate value. ;; How does one solve a function evaluating to indeterminate value.
use;limit;approaching;infinity
Use Limit of the function as n approaching infinity
no;method;solve;evaluating;0 by 0
there is no method to solve a function evaluating to 0 by 0
The answer is "Use Limit of the function as n approaching infinity."
A car is moving with speed given as a function of time. Approximate distance is given by a sum. When n equals infinity, the expression gives continuous-aggregate distance as 0 by 0 : indeterminate value. ;; Since the distance evaluates to indeterminate value, the limit is used to check if the function is defined. ;; When limit n tending to infinity is applied, the expression is computed. ;; at the end, The continuous-aggregate distance is computed as an algebraic expression.
A car is moving with speed given as function of time. The approximate distance is given by a sum. Continous aggregate distance traveled is given as limit of summation. Note that the distance traveled is computed starting from time t = 0, at which point the initial distance of the car can be non-zero. So the distance is given as initial distance at time 0, +, distance traveled between time 0 and t. ;; For the given problem distance = c + limit of sum, where c is a constant.
Summarizing the learning so far.;; Two quantities are in a cause-effect relation. ;; the cause is calculated as a function of an algebraic expression in a variable. ;; The effect is derived to be "aggregate of change of cause with respect to the variable. ;; In such a case, the effect is another algebraic expression in the variable. ;; The effect is computed as, the continuous-aggregate of change: the aggregate of the cause, over an interval, with the interval split into infinite partitions. ;; This calculation is named as integration or integral of the function.
Note 1: The summation in integral has many other forms, Riemann, Lebesgue, and Darboux forms, which will be introduced in due course. ;; Note 2: Differentiation is instantaneous rate of change, or, rate of change. Integration is continuous aggregate of change, or, aggregate-of-change.
Which of the following is a meaning for the word 'integrate'?
combine;1;with;another;form;whole
combine one with another to form a whole
convert;fraction;integer
convert a fraction to an integer
The answer is 'combine one with another to form a whole'. The aggregate of change is aptly named integral or integration.
What is the term used to refer "continuous-aggregate" of a function ?
integration;integrate
Integration in the context of cause-effect pair in continuous aggregate relation: If the cause is given by f of x then the effect is computed as integration or integral of f of x denoted as, integral f of x dx, = c +, limit n tending to infinity, sum from i=1 to n, f of i x by n, multiplied, x by n. ;; The second term is understood to be continuous aggregate between 0 and x. This is denoted as integral f of x dx, = c +, integral from 0 to x, f of x dx. = c + limit of summation.
Cause-effect was explained to understand the physical significance. Abstracting this and understanding the quantities involved in integration: A quantity u a function of x, is related to another quantity v, such that v is the aggregate of change of u with respect to x. then v =, integral u dx , = c + integral from 0 to x, u d x ;; = c +, limit n tending to infinity, sum from i=1 to n, f of i x by n, multiplied x by n. note that integral u d x is another quantity v, related to the given quantity u. ;; f of x is called the integrand. x is the variable of integration. c is the constant of integration. in integral from 0 to x, the value 0 denotes the start position of integration called lower limit of integration. in integral from 0 to x, the variable x denotes the end position of integration called upper limit of integration.
What is aggregate of change of a function called?
integration;integral
The answer is "integration or integral of the function".
Students can connect the notation integral f of x d x as ;; the small difference in x is given as d x. The multiplication by f of x to d x implies, the value of the function for the small difference is multiplied. The integral symbol denotes sum with limit of d x becoming close to 0. ;; Integral f of x d x = c + integral from 0 to x f of x d x. The left hand side give the general form of integration. The C, constant of integration is the initial value of the result. The 0 and x are the starting point and terminal points of the summation.
The aggregate of change of a function with respect to the variable is the integration of the function.
integral or integration of a function is defined.
Finding integral of y = 7x in first principles is given. What does the above prove.
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The answer is "integral 7 x dx = c + 7 x squared by 2"
Finding integral of y = sine x in first principles is given. Is it possible to simplify this resolve the indeterminate value 0 by 0 of limit?
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The answer is "all the above". Many functions are not easily solved by forward summation as defined by the first principles. We will learn properties of the limit of summation to solve integration of such functions.

we are not perfect yet...