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

1. Dec 27, 2008

### carlodelmundo

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:

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

Thanks

2. Dec 27, 2008

### Dick

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?

3. Dec 29, 2008

### carlodelmundo

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

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

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.

4. Dec 29, 2008

### Dick

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).

5. Dec 30, 2008

### carlodelmundo

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:

Is my answer -1/3 Units^2 / min correct?

6. Dec 30, 2008

### Dick

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

Looks good to me!