## BaoZhu Group Title 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 6 months ago 6 months ago

1. myininaya Group Title

May I ask what method of integration you chose?

2. BaoZhu Group Title

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

3. myininaya Group Title

you could have done this with just an algebraic substitution

4. myininaya Group Title

$u=16-x^2 => du=-2x dx$ $x^2=16-u$ 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 Group Title

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

6. BaoZhu Group Title

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

7. myininaya Group Title

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 Group Title

expression* that follows that first equal sign

9. myininaya Group Title

do you see?

10. BaoZhu Group Title

one more question

11. myininaya Group Title

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

12. BaoZhu Group Title

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

13. myininaya Group Title

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

14. myininaya Group Title

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 Group Title

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

16. myininaya Group Title

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

17. myininaya Group Title

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

18. myininaya Group Title

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

19. myininaya Group Title

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

20. myininaya Group Title

oh are those commas?

21. myininaya Group Title

lol

22. myininaya Group Title

23. myininaya Group Title

ok what you wrote is good then

24. myininaya Group Title

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

25. BaoZhu Group Title

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

26. BaoZhu Group Title

i don't understand

27. myininaya Group Title

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

28. myininaya Group Title

there is only one variable

29. myininaya Group Title

and you are asked to replace that variable x with 5.5

30. myininaya Group Title

this will give you P(5.5)

31. myininaya Group Title

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 Group Title

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

33. myininaya Group Title

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

34. BaoZhu Group Title

but now i got it, 105625 would be the answer

35. myininaya Group Title

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

36. myininaya Group Title

37. myininaya Group Title

don't forget the half part on that last term

38. myininaya Group Title

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

39. myininaya Group Title

which you wrote earlier has 22500 which is fine

40. myininaya Group Title

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