## anonymous 5 years ago find the area between the curves f(x)=(2-x)^2 and g(x)=square root of x for the interval [0,3]

1. TuringTest

we are going to need the point where these two graphs intersect, because they will cross each other, can you find that?

2. TuringTest

|dw:1327193397919:dw|we need the shaded area. Since area is always positive, we always need to subtract the lower area from the upper one, which meant we will need to split up our integral at the point where the graphs cross. To find that figure out when$f(x)=g(x)$

3. anonymous

I do but I don't get the part the area for the interval [0,3]. Like we need to find the area of the two portions, one from 0 to 1 and one from 1 to 3 or what????

4. TuringTest

yes those are the right intervals

5. anonymous

Use Riemann sums, Integration is overrated :P

6. TuringTest

so which interval will we want to use$\int f(x)-g(x)dx$and which will we use$\int g(x)-f(x)dx$?? @FFM I'm actually reviewing that now

7. anonymous

Great, which one do you like most the right or the left one?

8. TuringTest

their average :P

9. anonymous

hehe :D Can't we use there median? :P

10. anonymous

hey wait wait wait for the interval [0,1] we use $\int\limits_{0}^{1} f(x)-g(x)$ and [1,3] we use g(x)-f(x) right?

11. TuringTest

right :)

12. anonymous

Turing is a awesome teacher! :)

13. anonymous

@turing test.. your graph is not right... f(x)=(2-x)^2 have minima at x=2, and also x=2 is the solution... after that it may not cut squareroot of x before 3

14. TuringTest

thx, only to good students @sam I really was only trying to illustrate the intersection, there is no scale, so it flies for our purposes

15. TuringTest

(1,1) is the intersection

16. anonymous

do you know how to find that area by calculator?

17. TuringTest

no, I know how to find it by integrating

18. anonymous

how? I can't find the area for [1,3] by calculator

19. anonymous

YOu could use Mathematica :)

20. anonymous

Turning Test: thanks for your help. :) I'm trying to get the answer. FoolForMath: keke i know right. but it takes so long

21. TuringTest

$\int_{1}^{3}g(x)-f(x)dx=\int_{1}^{3}x^{1/2}-(2-x)^2dx=\int_{1}^{3}x^{1/2}-(4-4x+x^2)dx$got it from here?

22. TuringTest

$=\int_{1}^{3}x^{1/2}-(4-4x+x^2)dx=\int_{1}^{3}x^{1/2}-4+4x-x^2dx$...equals blah blah It's just algebra after integrating

23. anonymous

ok I'm confused now ~.~

24. TuringTest

where did you get stuck? did you understand everything I typed so far?

25. anonymous

you didn't type the one from [0,1), did you?

26. TuringTest

no, I though you only were having trouble with the second Is that just because you did the other with a calculator? lol ...I'll type out what we have so far while you respond.

27. anonymous

:) hey I got the answer but it's probably wrong. cuz I got the answer from my teacher. my answer is 2.46 and his answer is 3.797.

28. TuringTest

$\int_{0}^{1}f(x)-g(x)dx+\int_{1}^{3}g(x)-f(x)dx$$=\int_{0}^{1}(2-x)^2-x^{1/2}dx+\int_{1}^{3}x^{1/2}-(2-x)^2dx$$\int_{0}^{1}2-4x+x^2-x^{1/2}dx+\int_{1}^{3}x^{1/2}-4+4x-x^2dx$...

29. TuringTest

$=2x-2x^2+\frac{1}{3}x^3-\frac{2}{3}x^{3/2}|_{0}^{1}+\frac{2}{3}x^{3/2}-4x+2x^2-\frac{1}{3}x^3|_{1}^{3}$$=(2-2+\frac 1 3-\frac 2 3)+[\frac2 3(3\sqrt3)-4(3)+2(9)]-[\frac2 3 -4+2-\frac1 3]$...just simplifying from here...

30. anonymous

here is what i got after integrating $[4x- 2x ^{2}+ 1/3 x ^{3}-2/3x ^{3/2}]- [2/3x ^{3/2}-4x+2x ^{2}-1/3x ^{3}]$

31. TuringTest

that is for the first interval, right?

32. anonymous

no, it's for both. ahhhhhhh it's supposed to be + not -

33. anonymous

I'm sorry if it takes so long. sorry. you dont need to solve it though. I'll try to do it myself. :)

34. TuringTest

I am showing both at once here I made a typo above actually, fixed it here$\int_{0}^{1}f(x)-g(x)dx+\int_{1}^{3}g(x)-f(x)dx$$=\int_{0}^{1}(2-x)^2-x^{1/2}dx+\int_{1}^{3}x^{1/2}-(2-x)^2dx$$=\int_{0}^{1}4-4x+x^2-x^{1/2}dx+\int_{1}^{3}x^{1/2}-4+4x-x^2dx$$=4x-2x^2+\frac{1}{3}x^3-\frac{2}{3}x^{3/2}|_{0}^{1}+\frac{2}{3}x^{3/2}-4x+2x^2-\frac{1}{3}x^3|_{1}^{3}$$=(4-2+\frac 1 3-\frac 2 3)+[\frac2 3(3\sqrt3)-4(3)+2(9)]-[\frac2 3 -4+2-\frac1 3]$$=(2-\frac1 3)+2\sqrt3+6-(\frac1 3-2)=10-\frac2 3+2\sqrt3$I don't think that is the answer you wanted, so I must have made an arithmetic mistake somewhere. This is why I usually don't do the definite integration myself. Unless I happen to spot my mistake it will be your job to find it. I know my formulation is correct.

35. TuringTest

Oh I found the typo I think...

36. TuringTest

I dropped a term at the end$\int_{0}^{1}f(x)-g(x)dx+\int_{1}^{3}g(x)-f(x)dx$$=\int_{0}^{1}(2-x)^2-x^{1/2}dx+\int_{1}^{3}x^{1/2}-(2-x)^2dx$$=\int_{0}^{1}4-4x+x^2-x^{1/2}dx+\int_{1}^{3}x^{1/2}-4+4x-x^2dx$$=4x-2x^2+\frac{1}{3}x^3-\frac{2}{3}x^{3/2}|_{0}^{1}+\frac{2}{3}x^{3/2}-4x+2x^2-\frac{1}{3}x^3|_{1}^{3}$$=(4-2+\frac 1 3-\frac 2 3)+[\frac2 3(3\sqrt3)-4(3)+2(9)-9]-[\frac2 3 -4+2-\frac1 3]$$=(2-\frac1 3)+2\sqrt3-3-(\frac1 3-2)=1-\frac2 3+2\sqrt3$which is the answer stated by your teacher. You got lucky I found it, I was gonna make you do it ;)

37. anonymous

whoa you're a genius. thanks a lot. i'm trying to find my mistake here. thank you thank you

38. TuringTest

please do look for it, it's all about practice, I'm no genius your welcome, good luck :D

39. anonymous

yeyyyyyyyyyyyyyyyyyyyyyy i got it. Thanks TurningTest

40. TuringTest

very welcome!