Quantcast

A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

MATTW20

  • 2 years ago

Help Trigonometric Substitution \[\int\limits_{}^{}\sqrt{64-x ^{2}}\]

  • This Question is Closed
  1. MATTW20
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @hartnn

  2. SACAPUNTAS
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \[\cos^2 \theta = 1 - \sin^2 \theta\]

  3. MATTW20
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok

  4. SACAPUNTAS
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Do you see how that helps us here?

  5. MATTW20
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes because that will get you \[\sqrt{64\cos ^{2}\theta }\]

  6. MATTW20
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    and then dx=8cos t dt t for theta

  7. SACAPUNTAS
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Good. Keep going!

  8. MATTW20
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    now this is where i'm getting kind of lost

  9. SACAPUNTAS
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    The expression will be kind of messy, I think. You'll have to integrate it by parts. Do you know how to do that?

  10. MATTW20
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yeah we passed that section. I just am not super familiar with it. but could you just write the expression I'm supposed to integrate by parts so I make sure everything is correct up to that point

  11. MATTW20
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    and then i'll work on integrating that

  12. SACAPUNTAS
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Sure, one sec

  13. SACAPUNTAS
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Actually I think I spaced out a bit. I'm just getting \(\int d\theta\)

  14. MATTW20
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    no the answer is something weird \[\frac{ x }{ 2 }\sqrt{64-x ^{2}}+32\sin^{-1} (\frac{ x }{ 8 })+C\]

  15. SACAPUNTAS
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Ok, that looks like the result of integration by parts after all. So how did you get that? :D

  16. MATTW20
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    back of the book lol

  17. MATTW20
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    that's why I'm having trouble I can't get there

  18. myininaya
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    So what did you @MATTW20 let x=?

  19. myininaya
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Oh I see you let x=8sin(theta) and then you said dx=8cos(theta) d theta Just plug into your expression.

  20. myininaya
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \[\int\limits_{}^{}\sqrt{64\cos^2(\theta) } 8 \cos(\theta) d \theta \] That is what you have got so far, right?

  21. MATTW20
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    correct

  22. myininaya
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \[\int\limits_{}^{}8 \cos(\theta) \cos(\theta) d \theta ,\text{ assuming } \cos(\theta)>0\] so we have that we are trying to evaluate: \[8\int\limits_{}^{}\cos^2(\theta) d \theta \] The double angle identity comes in use for integating cos^2(theta) and even sin^2(theta)

  23. surjithayer
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \[put x=8\sin \theta ,dx=8\cos \theta d \theta \] \[I=\int\limits \sqrt{64-64\sin ^{2}\theta} 8\cos \theta d \theta =32\int\limits 2\cos ^{2}\theta d \theta\] \[=32\int\limits \left( 1+\cos 2\theta \right)d \theta=32\left( \theta+\frac{ \sin 2\theta }{2 } \right)+c\] |dw:1381785998999:dw| \[I=32\left( \theta+\frac{ 2\sin \theta \cos \theta }{ 2 } \right)+c\] \[I=32\left( \sin^{-1} \frac{ x }{8 }+\frac{ x }{8 }*\cos \frac{ \sqrt{64-x ^{2}} }{8 } \right)+c\]

  24. myininaya
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \[\cos(2\theta)=\cos^2(\theta)-\sin^2(\theta)=\cos^2(\theta)-(1-\cos^2(\theta))\] \[\cos(\theta)=\cos^2(\theta)-1+\cos^2(\theta)=2\cos^2(\theta)-1\] Solve for cos^2(theta) This is just the way I remember this formula. I remember other formulas and derive other formulas from the ones I have memorized. \[\frac{1}{2}(\cos(\theta)+1)=\cos^2(\theta) \]

  25. MATTW20
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok

  26. myininaya
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    And yes I forgot to write the other 8.. Errr...

  27. zepdrix
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\Large \color{teal}{\cos^2(\theta)\quad=\quad \frac{1}{2}(\cos2\theta+1)}\]Woops! Careful with your half-angle formula there missy! c:

  28. myininaya
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Yeah I missed my 2.

  29. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.