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

  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. Dick

    Dick 25,664
    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. 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. Dick

    Dick 25,664
    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. 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. Dick

    Dick 25,664
    Science Advisor
    Homework Helper

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

    Looks good to me!
     
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