## anonymous 5 years ago 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

1. anonymous

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

2. anonymous

i already checked that, i need better explaination

3. anonymous

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

4. anonymous

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

5. anonymous

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

6. anonymous

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

7. anonymous

are you familiar with riemann sums?

8. anonymous

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

9. anonymous

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

10. anonymous

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

11. anonymous

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

12. anonymous

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

13. anonymous

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

14. anonymous

There you go

15. anonymous

does it makes sense?

16. anonymous

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

17. anonymous

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

18. anonymous

no problem, I'm glad you understand

19. anonymous

Are you in highschool?

20. anonymous

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

21. anonymous

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)

22. anonymous

cool

23. anonymous

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

24. anonymous

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

25. anonymous

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

26. anonymous

For part 2

27. anonymous

Thank you.