## BaoZhu one year ago integral x^3(sqrt(16-x^2)dx can someone check if my answer is right, (-1024/3)((sqrt(16-x^2)/(3)))^3+(1024/5)((sqrt(16-x^2)/(3)))^5+C

1. myininaya

May I ask what method of integration you chose?

2. BaoZhu

first i used trig. sub. a-x^2, gives x=4sintheta, and then i used trig. integral of u-sub.

3. myininaya

you could have done this with just an algebraic substitution

4. myininaya

$u=16-x^2 => du=-2x dx$ $x^2=16-u$ So $\int\limits_{}^{}x^2 \sqrt{16-x^2} dx=\frac{-1}{2}\int\limits_{}^{}x^2 \sqrt{16-x^2} (-2x) dx$ $=\frac{-1}{2}\int\limits_{}^{}(16-u) \sqrt{u} du$ $=\frac{-1}{2}\int\limits_{}^{}(16 u^\frac{1}{2}-u^\frac{3}{2}) du$

5. BaoZhu

but it's x^3, not x^2

6. BaoZhu

integral x^3*sqrt(16-x^2)dx

7. myininaya

I missed an x in that first part but and that next equation you see i have x^2 times x which is x^3

8. myininaya

expression* that follows that first equal sign

9. myininaya

do you see?

10. BaoZhu

one more question

11. myininaya

Did you find P(x) as a 2nd degree taylor polynomial yet?

12. BaoZhu

i do not know where to start......

13. myininaya

$P(x) \approx P(r)+P'(r)(x-r)+\frac{1}{2}P''(r)(x-r)^2$ This is what they want you to use.

14. myininaya

They want you to use r as 5 and plug in the values they gave you for P'(5) and P''(5) Then they want you to find P(5.5) approximately using the resulting equation

15. BaoZhu

so for 5, i got p5(x)=120000-40000(x-5)+22500(x-5)^2, is this right so far?

16. myininaya

$P(x)=45000+\frac{45000}{120000}(x-5)+\frac{1}{2} \frac{45000}{-40000}(x-5)^2$

17. myininaya

We are given $P(5)=45000; \text{ and } 120000 \cdot P'(5)=45000 ; \text{ and } -40000 \cdot P''(5)=45000$

18. myininaya

That is how I got what was P'(5) and P''(5) and P(5)

19. myininaya

We will use that P I wrote to approximate the value of portfolio bonds when r is 5.5.

20. myininaya

oh are those commas?

21. myininaya

lol

22. myininaya

23. myininaya

ok what you wrote is good then

24. myininaya

So use your equation not mine to approximate P(5.5)

25. BaoZhu

but how will it turn the equation into just 1 number?solve for x?

26. BaoZhu

i don't understand

27. myininaya

$P(x) \approx 120000-40000(x-5)+45000(x-5)^2$ You can find P(5.5) It will result in one number

28. myininaya

there is only one variable

29. myininaya

and you are asked to replace that variable x with 5.5

30. myininaya

this will give you P(5.5)

31. myininaya

It is just like if i asked you to evaluate f(2) given f(x)=x-5 you would say f(2)=2-5=-3

32. BaoZhu

oh thank you so much, i thought i have to replace 5.5 in as in term of r

33. myininaya

and don't forget the 1/2 part on that one part

34. BaoZhu

but now i got it, 105625 would be the answer

35. myininaya

yeah but you aren't given P''(5.5) or P'(5.5) or P(5.5) so that would be impossible

36. myininaya

37. myininaya

don't forget the half part on that last term

38. myininaya

$P(x) \approx 120000-40000(x-5)+\frac{1}{2} 45000(x-5)^2$

39. myininaya

which you wrote earlier has 22500 which is fine

40. myininaya

I'm talking about for that 1/2*45000 part you simplified it earlier to 22500