Area between two curves problem

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In summary, the conversation involves finding the area A as a function of j, where j is the value used to define the region in the 1st quadrant enclosed by the y-axis and the graphs of y = x ^ (1/3) and y = j. The value of A is 4 when j = 2, and the rate of change of A is zero when the line y = k is moving upward at a rate of 1/2 units per second.
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Homework Statement



(a) Find the area A, as a function of j, of the region in the 1st quadrant enclosed by the y-axis and the graphs of y = x ^ (1/3) and y = j for j > 0.

(b) What is the value of A when k = 2

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



2. The attempt at a solution

Ok, so I doubt this is right as I was really, confused, but this is what I did...

(a)
x^(1/3) = j
x = j^3

A(j) = integral of (j - x^(1/3))dx from 0 to j^3

(b) A(2) = integral of (2 - x^(1/3))dx from 0 to 2^3
this equals = 4

(c) I have no idea..
I took the derivative of A(x) and got zero...

Please help. :)
 
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  • #2
You computed corretcly the Area, which is a function of ? (name it)

If j changes, and you have a certain function of j, how do you compute the change of f(j) ?
 

Related to Area between two curves problem

What is the "area between two curves" problem?

The "area between two curves" problem is a mathematical concept that involves finding the area of the region between two curves on a graph. This can be done by finding the definite integral of the difference between the two curves over a given interval.

What are the steps to solve the "area between two curves" problem?

The steps to solve the "area between two curves" problem are:

  1. Graph the two curves on the same coordinate plane.
  2. Determine the points of intersection between the two curves.
  3. Set up the definite integral by subtracting the lower curve from the upper curve.
  4. Integrate the equation over the interval determined by the points of intersection.
  5. Simplify and calculate the area between the curves.

Can the "area between two curves" problem be solved without calculus?

No, the "area between two curves" problem requires the use of calculus to find the exact area. Other methods, such as using geometric formulas or approximations, may provide an estimate but they will not give an exact answer.

What is the significance of the "area between two curves" problem?

The "area between two curves" problem has practical applications in various fields such as engineering, physics, and economics. It allows us to calculate the area of irregular shapes and can be used to solve optimization problems.

Are there any common mistakes to avoid when solving the "area between two curves" problem?

One common mistake is to forget to subtract the lower curve from the upper curve when setting up the definite integral. Another mistake is to integrate the wrong equation or over the wrong interval. It is important to carefully follow the steps and double-check your work to avoid these errors.

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