Question

Theory: If f and g are continuous functions on the interval [a, b], and if f(x) ≥ g(x) for all x in [a, b], then the area of the region bounded above by y = f(x), below by y = g(x), on the left by the line x = a, and on the right by the line x = b is

b
A = ∫ [f(x) – g(x)] dx
a

Explain the theory and why it is true

* 20 hours ago
* - 3 days left to answer.

Additional Details
This question will be for a research paper. in essay format

Answer 1

http://tutorial.math.lamar.edu/cheat_table.aspx
go to the complete cheat sheet for calculus, its explained there

Answer 2

i already checked that, i need better explaination

Answer 3

Well think of it this was if you have a function f(x) whose values are greater than a function g(x) then its only true that the function f(x) lies above g(x) for all values of that function. So in order to determine the area bounded by both function you would have to subtract greater function f(x) from the lesser function g(x) and integrate to determine the area contained within these two functions

Answer 4

in essence these two functions function as boundaries for the area contained within them

Answer 5

well, what im trying to explain is, why would you intrgral or (antiderive it)? can you explain that aswell/.

Answer 6

aka = The Fundamental Theorems of the Calculus ( 1 and 2)

Answer 7

are you familiar with riemann sums?

Answer 8

Do you under stand the concept of this limit
\[f'(a)=\lim_{h \rightarrow 0} [f(a+h)-f(a)]/h\]

Answer 9

Riemann sums are were integrals come from and goes hand in hand with the limit I posted above which coincides with the FTC 1 and 2

Answer 10

If you think about the limit I posted above it explains the the derivative but fails to exist at h=o, but the idea to get as close to 0 without ever being 0 which give you the derivative. Riemann sums work the same way but instead it uses smaller and smaller rectangles and approaches the function to determine the area under the function which is multiplied by the minute changes in x hence that's where you always notice a dx in integral: hence base x height which give you the area

Answer 11

yes i understand the riemann sum, it adds up all the boxes within the interval, to find the total area accordin to your limit

Answer 12

and the smaller you make the boxes the closer you get to the function

Answer 13

height = f(x) , base = dx , ami rite?

Answer 14

There you go

Answer 15

does it makes sense?

Answer 16

As f(x) changes to smaller and smaller intervals do does dx hence you have the change in y over the change in x which is a derivative....... multiplied over the entire function, which is the area

Answer 17

yea makes sense now, thank you, i will be sure to ask more questions if i need anymore help. THANK YOU!

Answer 18

no problem, I'm glad you understand

Answer 19

Are you in highschool?

Answer 20

I only ask bc I'm in college and I never had to prove such a thing for any of my cal classes

Answer 21

nope in uni. just a conus question for my midterm, i have to write an essay during my mid!! (counts only as bonus points towards mid marks)

Answer 22

cool

Answer 23

can u just explain the second FTC, how it works?

Answer 24

Part 1 says
\[\int\limits_{a}^{b}f(x)dx= F(b)-F(a)\]
Where F'=f
Part 2 says
\[d/dx(\int\limits_{a}^{x}f(t)dt)=f(x)\]

Answer 25

http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
Wikipedia has an example of it under the Example section take a look at it and see if you can understand the concept

Answer 26

For part 2

Answer 27

Thank you.