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

In summary, the conversation discusses finding the area of a region in the first quadrant under the graph of y = cos(x) and above the line y = k, using the Fundamental Theorem of Calculus. The conversation also addresses determining the value of A for a specific value of k, and finding the rate of change of the area when the line y = k is moving upward at a given rate. The solution involves correctly finding the derivative of A with respect to k, and then using the chain rule to find the rate of change of A with respect to time. The final answer is -1/3 Units^2 / min.
  • #1
carlodelmundo
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0

Homework Statement



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 ?

Homework Equations




Fundamental Theorem of Calculus

The Attempt at a Solution



Here's my work insofar:

http://carlodm.com/calc/prob2.jpg [Broken]

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

Thanks
 
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  • #2


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


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

http://carlodm.com/calc/prob4.JPG [Broken]

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.
 
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  • #4


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


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 [Broken]

Is my answer -1/3 Units^2 / min correct?
 
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  • #6


Looks good to me!
 

1. What is the general process for solving integration problems involving constants of integration?

The general process for solving integration problems involving constants of integration is as follows:

  • Step 1: Identify the given function and its corresponding derivative.
  • Step 2: Rewrite the function in terms of its derivative and a constant of integration.
  • Step 3: Integrate the function with respect to the independent variable.
  • Step 4: Solve for the constant of integration using any additional given information or boundary conditions.
  • Step 5: Write the final solution with the constant of integration included.

2. How do I determine the appropriate constant of integration to use in an integration problem?

The constant of integration can be determined by using any additional given information or boundary conditions in the problem. These can include initial values, limits, or other known values of the function or its derivative. The constant of integration should be chosen such that the final solution satisfies these conditions.

3. Can I use the constant of integration to solve for any unknown variables in the problem?

No, the constant of integration is only used to account for the unknown constant in the indefinite integral. It cannot be used to solve for other unknown variables in the problem. Additional information or equations are needed to solve for other variables.

4. How are related rates involved in integration problems?

Related rates refer to the relationship between the rate of change of a function and its independent variable. In integration problems, related rates are used to find the rate of change of a function when its derivative is known. This can be done by taking the derivative of the function with respect to time and using the chain rule to solve for the related rate.

5. What is the difference between definite and indefinite integration?

Definite integration involves finding the exact numerical value of an integral between given limits. This results in a specific numerical answer. Indefinite integration, on the other hand, involves finding the general expression for the antiderivative of a function. This results in a solution with a constant of integration, which can be used to find the specific solution to a definite integral.

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