Results 1 to 8 of 8

Math Help - A few Calc II Final Exam Questions (Setup ONLY)

  1. #1
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1

    A few Calc II Final Exam Questions (Setup ONLY)

    Just a few setup problems... If I knew what strategy to use to evaluate these integrals, I'd be better off. That's all I need to know for the following:

    1.) Find the volume of the solid formed by revolving the region bounded by the graphs of y = -x^2 + 4 and y = 0 about the x-axis.

    2.) Evaluate:


    \int_{-4}^4 \sqrt{16 - x^2}dx

    3.) Evaluate:

    \int \ln{2x} \ dx
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Apr 2008
    Posts
    1,092
    Quote Originally Posted by Aryth View Post
    Just a few setup problems... If I knew what strategy to use to evaluate these integrals, I'd be better off. That's all I need to know for the following:

    1.) Find the volume of the solid formed by revolving the region bounded by the graphs of y = -x^2 + 4 and y = 0 about the x-axis.

    2.) Evaluate:

    \int_{-4}^4 \sqrt{16 - x^2}dx

    3.) Evaluate:

    \int \ln{2x} \ dx
    For the first problem, use the disc method and integrate:
    \int_{-2}^{2} \pi{(4 - x^2)^2} dx
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1
    Thanks, that helped tremendously.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    o_O
    o_O is offline
    Primero Espada
    o_O's Avatar
    Joined
    Mar 2008
    From
    Canada
    Posts
    1,407
    2. y = + \sqrt{16 - x^{2}}
    y^{2} = 16 - x^{2}
    x^{2} + y^{2} = 16

    The integral represents the area under the upper half of a circle of radius 4. So: \text{A} = \frac{1}{2}\pi r^{2}

    3. u = 2x \: \Rightarrow \: du = 2 dx \: \Rightarrow \: dx = \frac{du}{2}

    Making the subs:
    \int \ln u \: \frac{du}{2} = \frac{1}{2} \int \ln u \: du
    which is a standard one
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Apr 2008
    Posts
    1,092
    I'm not sure about #2, but for #3 the antiderivative of \ln{x} is x \ln{x} - x + C.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1
    Quote Originally Posted by o_O View Post
    2. y = + \sqrt{16 - x^{2}}
    y^{2} = 16 - x^{2}
    x^{2} + y^{2} = 16

    The integral represents the area under the upper half of a circle of radius 4. So: \text{A} = \frac{1}{2}\pi r^{2}
    Wow, thanks. That was completely different then what I was expecting. But the answer was correct.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Apr 2008
    Posts
    1,092
    Quote Originally Posted by Aryth View Post
    Wow, thanks. That was completely different then what I was expecting. But the answer was correct.
    Yeah, I tried using trig substitution on that one. Not the way to go about it, apparently.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    o_O
    o_O is offline
    Primero Espada
    o_O's Avatar
    Joined
    Mar 2008
    From
    Canada
    Posts
    1,407
    Well you could do a trig sub:
    x = 4\sin \theta
    dx = 4\cos \theta d\theta

    So you get:
    = \int \sqrt{16 - 16\sin^{2} \theta} \cdot 4\cos \theta d\theta
    = 4 \int \sqrt{16} \sqrt{1 - \sin^{2} \theta} \cos \theta d\theta

    etc. etc.

    But as you can see having that little insight with recognizing the form of a circle speeds up the process.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Final Exam Help:/ ?
    Posted in the Algebra Forum
    Replies: 2
    Last Post: May 25th 2010, 05:17 PM
  2. Final exam tomorrow, need help!
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 4th 2010, 06:11 PM
  3. Help Preparing for Final Exam
    Posted in the Number Theory Forum
    Replies: 19
    Last Post: August 4th 2009, 01:11 PM
  4. Final Exam Review - Help
    Posted in the Algebra Forum
    Replies: 5
    Last Post: May 10th 2009, 08:45 PM
  5. final exam help >.<
    Posted in the Math Topics Forum
    Replies: 6
    Last Post: May 3rd 2007, 03:15 PM

Search Tags


/mathhelpforum @mathhelpforum