Calc BC  Integration Problem involving Constants of Integration and Related Rates
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: http://carlodm.com/calc/prob2.jpg For c.) I have no idea on how to tackle this problem. Should I derive my area formula in terms of dt? Thanks 
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?

Re: Calc BC  Integration Problem involving Constants of Integration and Related Rate
Thank you Dick for your help. Here's my new work:
http://carlodm.com/calc/prob4.JPG 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. 
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).

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: http://carlodm.com/calc/prob6.PNG Is my answer 1/3 Units^2 / min correct? 
Re: Calc BC  Integration Problem involving Constants of Integration and Related Rate
Looks good to me!

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