1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Calc BC - Integration Problem involving Constants of Integration and Related Rates

  1. Dec 27, 2008 #1
    1. The problem statement, all variables and given/known data

    Let A be the area of the region in the first quadrant under the graph of y = cos (x) and above the line y = k for 0 <= k <= 1.

    a.) Determine A in terms of k.

    b) Determine the value of A when k = 1/2.

    c) If the line y = k is moving upward at the rate of ( 1 / pi ) units per minute, at what rate is the area, A, changing when k = 1/2 ?

    2. Relevant equations


    Fundamental Theorem of Calculus

    3. The attempt at a solution

    Here's my work insofar:

    [​IMG]

    For c.) I have no idea on how to tackle this problem. Should I derive my area formula in terms of dt?

    Thanks
     
  2. jcsd
  3. Dec 27, 2008 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Calc BC - Integration Problem involving Constants of Integration and Related Rate

    It would really help if you could make a smaller scan. You are doing fine up to c). First you need to find dA/dk correctly. A(k)=sin(arccos(k))-k*arccos(k). It looks like its almost right, except why are you mixing x's and k's. Shouldn't they all be k's?
     
  4. Dec 29, 2008 #3
    Re: Calc BC - Integration Problem involving Constants of Integration and Related Rate

    Thank you Dick for your help. Here's my new work:

    [​IMG]

    I have a problem. I have the rate of change of area with respect to time in terms of x. I'm given a rate of change in terms of y. I thought to myself that maybe I can just "cheat" and plug in dK/dt for dx/dt, but isn't this wrong?

    Basically, can you give me tips to solve for the rate of change of y? I'm stumped.
     
  5. Dec 29, 2008 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Calc BC - Integration Problem involving Constants of Integration and Related Rate

    You are sort of confusing x and k. x is the variable you are integrating over. The upper limit is arccos(k). Area should just come out as a function of k. I get that dA/dk=-arccos(k). dA/dt=dA/dk*(dk/dt).
     
  6. Dec 30, 2008 #5
    Re: Calc BC - Integration Problem involving Constants of Integration and Related Rate

    Thanks a lot, Dick!

    You're right about me confusing x and k. I thought all related rates problems derived in terms of t, but now I can see that that's not always the case. That is a very elegant solution in my opinion.

    Here is my revised work:

    [​IMG]

    Is my answer -1/3 Units^2 / min correct?
     
  7. Dec 30, 2008 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Calc BC - Integration Problem involving Constants of Integration and Related Rate

    Looks good to me!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Calc BC - Integration Problem involving Constants of Integration and Related Rates
Loading...