Ilm Gah Website is under maintenance! If you found any error please let us know. Contact Us ×

Definite Integrals

SHARE:

You might like to read  Introduction to Integration  first! Integration Integration  can be used to find areas, volumes, central ...


You might like to read Introduction to Integration first!

Integration

Integration can be used to find areas, volumes, central points and many useful things. But it is often used to find the area under the graph of a function like this:
integral area
The area can be found by adding slices that approach zero in width:
And there are Rules of Integration that help us get the answer.
integral area dx

Notation

The symbol for "Integral" is a stylish "S"
(for "Sum", the idea of summing slices):
integral notation
After the Integral Symbol we put the function we want to find the integral of (called the Integrand),
and then finish with dx to mean the slices go in the x direction (and approach zero in width).

Definite Integral

Definite Integral has start and end values: in other words there is an interval (a to b).
The values are put at the bottom and top of the "S", like this:
indefinite integraldefinite integral
Indefinite Integral
(no specific values)
Definite Integral
(from a to b)
We can find the Definite Integral by calculating the Indefinite Integral at points a and b, then subtracting:
definite integral y=2x from 1 to 2 as graph

Example:

The Definite Integral, from 1 to 2, of 2x dx:
definite integral 2x dx from 1 to 2

The Indefinite Integral is: 2x dx = x2 + C
  • At x=1: 2x dx = 12 + C
  • At x=2: 2x dx = 22 + C
Subtract:
(22 + C) − (12 + C)
22 + C − 12 − C
4 − 1 + C − C = 3
And "C" gets cancelled out ... so with Definite Integrals we can ignore C.
In fact we can give the answer directly like this:
definite integral 2x dx from 1 to 2 = 2^2 - 1^2 = 3

area of y=2x from 1 to 2 equals 3
We can check that, by calculating the area of the shape:
Yes, it has an area of 3.
(Yay!)
Let's try another example:
definite integral y=cos(x) from 0.5 to 1 graph

Example:

The Definite Integral, from 0.5 to 1.0, of cos(x) dx:
definite integral cos(x) dx from 0.5 to 1
(Note: x must be in radians)

The Indefinite Integral is: cos(x) dx = sin(x) + C
We can ignore C when we do the subtraction (as we saw above):
definite integral cos(x) dx from 0.5 to 1= sin(1) − sin(0.5)
= 0.841... − 0.479...
0.362...
And another example to make an important point:
definite integral y=sin(x) from 0 to 1 graph

Example:

The Definite Integral, from 0 to 1, of sin(x) dx:
definite integral sin(x) dx from 0 to 1

The Indefinite Integral is: sin(x) dx = −cos(x) + C
Since we are going from 0, can we just calculate the area at x=1?
−cos(1) = −0.540...
What? The Area at x=1 is negative? No, we need to subtract the integral at x=0. We shouldn't assume that it is zero.
So let us do it properly, subtracting one from the other (and C gets cancelled so we don't need to show it):
definite integral sin(x) dx from 0 to 1= −cos(1) − (−cos(0))
= −0.540... − (−1)
0.460...
That's better!
But we can have negative areas, when the curve is below the axis:
definite integral y=cos(x) from 1 to 3

Example:

The Definite Integral, from 1 to 3, of cos(x) dx:
definite integral cos(x) dx from 1 to 3
Notice that some of it is positive, and some negative.
The definite integral will work out the net area.

The Indefinite Integral is:cos(x) dx = sin(x) + C
So let us do the calculations:
definite integral cos(x) dx from 1 to 3= sin(3) − sin(1)
= 0.141... − 0.841...
−0.700...
Try integrating cos(x) with different start and end values to see for yourself how positives and negatives work.
But sometimes you want the actual area (without the part below being subtracted):
area y=cos(x) from 1 to 3 positive both above and below

Example: What is the area between y = cos(x) and the x-axis from x = 1 to x = 3?

This is like the example we just did, but area is positive (imagine you had to paint it).
So now we have to do the parts separately:
  • One for the area above the x-axis
  • One for the area below the x-axis
The curve crosses the x-axis at x = π/2 so we have:
    π/2
1

cos(x) dx = sin(π/2) − sin(1)
      = 1 − 0.841...
= 0.159...
    3
π/2

cos(x) dx = sin(3) − sin(π/2)
      = 0.141... − 1
  = −0.859...
That last one comes out negative, but we want positive, so:
Total area = 0.159... + 0.859... = 1.018...
This is very different from the answer in the previous example.

Continuous

Oh yes, the function we are integrating must be Continuous between a and b: no holes, jumps or vertical asymptotes (where the function heads up/down towards infinity).
not continuous asymptote

Example:

A vertical asymptote between a and b affects the definite integral.

Properties

Reversing the interval

definite integral negative property
Reversing the direction of the interval gives the negative of the original direction.
definite integral a to b = negative of b to a

Interval of zero length

definite integral area zero
When the interval starts and ends at the same place, the result is zero:
definite integral a to a = 0

Adding intervals

area a to b = a to c plus c to b
We can also add two adjacent intervals together:
definite integral a to b = a to c plus c to b

Summary

The Definite Integral between a and b is the Indefinite Integral at b minus the Indefinite Integral at a.

COMMENTS

Name

11 August 2017,1,21 July 2017,1,25 July 2017,1,Admissions,6,Agri-Jobs,1,Animated Moral Stories,2,Atomic-Jobs,1,Colleges,1,Computer,32,Conferences,2,CSS Agriculture,3,CSS Current Affairs,4,CSS Data,62,CSS Date Sheet,1,CSS English Essay,10,CSS Everyday Science,7,CSS General Knowledge,1,CSS History of Pakistan,2,CSS Islamiyat,4,CSS Pakistan Affairs,28,CSS Past Paper,59,Data,5,Date Sheets,5,Derivative,2,Differentiate between,2,Differentiation,1,Employees,1,English1 FSc Notes,15,English3 FSc Notes,3,English4 FSc Notes,2,Entry Test,1,Essay Writing,3,Exchange Programs,3,Food Jobs,2,Form of Verbs,26,Guess 2nd Year,10,Important Questions Chemistry2 FSc,4,Important Questions Physics2 FSc,5,Integration,5,Inter Date Sheet,1,Internships,2,Jobs,14,Moral-Stories,1,NAB-Jobs,1,News-Updates,6,Novel,1,OP Questions Chemistry2 FSc,4,OP Questions Physics2 FSc,5,Presentations,2,Quotes,1,Results,12,Roll Number Slips,1,Scholarships,15,Statistics,6,Test Chemistry2 FSc,1,Tips,3,Urdu Essays,1,Vacancies,1,Videos,2,Writting Tips,1,
ltr
item
Ilm Gah: Definite Integrals
Definite Integrals
https://3.bp.blogspot.com/-T46T6tw8Zac/WGo3kUKrQEI/AAAAAAAADSk/AAhFGV3v6X0Ja9bLeEJIUDRdGa8DrLiTwCLcB/s320/Integral_example.svg.png
https://3.bp.blogspot.com/-T46T6tw8Zac/WGo3kUKrQEI/AAAAAAAADSk/AAhFGV3v6X0Ja9bLeEJIUDRdGa8DrLiTwCLcB/s72-c/Integral_example.svg.png
Ilm Gah
https://www.ilmgah.com/2017/01/definite-integrals.html
https://www.ilmgah.com/
https://www.ilmgah.com/
https://www.ilmgah.com/2017/01/definite-integrals.html
true
5949122430306054992
UTF-8
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS PREMIUM CONTENT IS LOCKED STEP 1: Share to a social network STEP 2: Click the link on your social network Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy