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mathsIntegral CalculusAlgebra of Integrals

Properties of Definite Integrals

In this page, the properties of definite integrals, specifically based on the limits of integration, are explained.



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Definite integral is defined as `int_a^b f(x)dx = g(b)-g(a)`, where `g(x)` is found by either one of the following

 • aggregate-of-change represented by `lim_(n->oo) sum_(i=1)^n ``f(a+(i (b-a))/n) xx (b-a)/n`

 • the anti-derivative of `f(x)` represented by `d/(dx) g(x) = f(x)`.

 • the area under the curve `f(x)` between `a` and `b`.

To understand the properties of definite integrals one of these is used.

 • aggregate of change provides rigorous mathematical proof.

 • anti-derivatives provide proof derived from properties of derivatives.

 • area under the curve provide the geometrical methods.

A proof given in one of these three can be verified in other.

It is noted that anti-derivatives or indefinite integrals can be used to compute definite integrals.

The properties pertaining to addition, subtraction, and multiplication by constant are studied for indefinite integrals. Will those be applicable to definite integrals?

  • Yes, the properties are directly applicable
  • Yes, the properties are directly applicable
  • No, these have to be figured out independently for definite integrals

The answer is "Yes, the properties are directly applicable".

Consider two forms of definite integrals.

`int_a^b f(x)dx = g(b)-g(a)` and `int_a^b f(y)dy = g(b)-g(a)`.

The left hand side has different variables `x` and `y` inside the definite integrals. The right hand side is independent of the variables. What would be the value of the result of the two integrals? Will they be equal?

  • Yes, the results would be equal
  • Yes, the results would be equal
  • no, the results would not be equal

The answer is "Yes, the results would be equal".

Intuitive understanding of
`int_a^b f(x) dx = int_a^b f(y) dy`

In `int_a^b f(y)dy` the variable of integration is `y` and so the limits `a` to `b` are for `y`.

In `int_a^bf(x)dx` the variable of integration is `x` and so the limits `a` to `b` are for `x`.

Despite any relationship between `y` and `x`, the given definite integrals are identical.

Solved Exercise Problem:

Given `int_0^2 f(x)dx = 20` and `y=2x`, what is `int_0^2 f(y)dy` ?

  • `40`
  • `20`
  • `20`

The answer is "`20`".

Given `int_a^b f(x) dx = g(b)-g(a)`, what is the value of `int_a^a f(x) dx`?

  • `0`
  • `0`
  • `2g(a)`

The answer is "`0`". `int_a^a f(x)` `=g(a)-g(a)``=0`.

Intuitive understanding of
`int_a^a f(x) dx = 0` : The area under the curve is 0 between `a` and `a` as the width is `0`.

Given `int_a^b f(x)dx = k`, what is `int_b^a f(x)dx`?

  • `k`
  • `-k`
  • `-k`

The answer is "`-k`"

Intuitive understanding of
`int_a^b f(x) dx = -int_b^a f(x)dx` : The area under the curve is positive when the traversal is in increasing `x` direction. And the area under the curve is negative when the traversal is in the decreasing `x` direction. So, when the limits are switched, the direction of traversal reverses.

Given `int_a^b f(x)dx = k` and `int_b^c f(x)dx = l`, what is `int_a^c f(x)dx`? Note: `int_a^b f(x)dx` is the area under the curve from `a` to `b`. `int_b^c f(x)dx` is the area under the curve from `b` to `c`. The question is to find area under the curve from `a` to `c`.

  • `k+l`
  • `k+l`
  • cannot be computed

The answer is "`k+l`". That is `int_a^c f(x)dx` `=int_a^b f(x)dx` `+ int_b^c f(x)dx`.

Intuitive understanding of
`int_a^b f(x)dx` `+ int_b^c f(x)dx=``int_a^c f(x)dx`

 • the aggregate of change continues from the end of one integral to the second integral.continuing limits of integral  • Area under the curve adds under the given integrals.

Properties of definite integrals is understood as area under the curve

 • area under `a` to `b` is negative of area under `b` to `a`

 • area under `a` to `c` is sum of area under `a` to `b` and area under `b` to `c`

Properties of Definite Integrals:
`int_a^b f(x) dx = int_a^b f(y) dy`

`int_a^a f(x) dx = 0`

`int_a^b f(x) dx = -int_b^a f(x)dx`

`int_a^b f(x)dx` `+ int_b^c f(x)dx=``int_a^c f(x)dx`

Solved Exercise Problem:

Given
`int_a^b f(x) dx = 2` and `int_c^b f(x) dx = 1`, what is `int_a^c f(x)dx`?

  • `1`
  • `1`
  • `3`

The answer is "`1`".

`int_a^c f(x)dx`

`= int_a^b f(x)dx ``+int_b^c f(x)dx`

`= int_a^b f(x)dx ``-int_c^b f(x)dx`

`= 2-1`

                            
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