Area Under a Curve With a Constant, k

[carlodelmundo]

Homework Statement

Let f be the function given by f(x) = kx^2 - x^3, where k is a positive constant. Let R be the region in the first quadrant bounded by the graph of f and the x-axis.

Find all values of the constant k for which the area of R equals 2.

The Attempt at a Solution

So I've pretty much decided that we're comparing two terms here: the kx^2 and the x^3 term. If the x^3 term is bigger than the former, than voila we have our limits of integration!

I understand that squaring a number between 0 and 1 will make a smaller number... and cubing that same number makes an even smaller number. so, from 0 < x < 1, the x^2 term is bigger than the x^3.

I tried integrating the function f(x) from 0 to 1 and got a k value of 27/4. This leads me to think that there are infinite limits from x = 0 to x = b where I can get a region of 2.

Hints please

Thanks

 

[Avodyne]

Leave k as an arbitrary constant for now. What is the area of the region R? Your answer will be a function of k. Set that function equal to 2, and solve for k.

 

[Feldoh]

Your limits of integration are not correct. It's bounded in the first quadrant so find where the function is in the first quadrant. Hint: Look at the roots, and why is know about cubics.

 

[carlodelmundo]

Leave k as an arbitrary constant for now. What is the area of the region R? Your answer will be a function of k. Set that function equal to 2, and solve for k.

If I leave k as an arbitrary constant, I get the following:

integration of f(x) = 2

(k/3) x^3 - (1/4) x^4 + C = 2

The "C" value is throwing me off. I can't integrate correctly without the limits of integration. The limits of integration depend on the value k.

Your limits of integration are not correct. It's bounded in the first quadrant so find where the function is in the first quadrant. Hint: Look at the roots, and why is know about cubics.

I understand that my limits are wrong. I'm saying that depending on the value of k, the limits of integration change from 0 to b, where b is some positive number. I'm looking at the roots kx^2 and x^3 but i can't ascertain any other information...

Thanks for the help so far but I'm still a little stumped

 

[Mark44]

Your integrand is kx^2 - x^3. What are the x-intercepts of this function? Those will be your limits of integration. Your integral is a definite integral, so the constant C is unnecessary.

 

[carlodelmundo]

Your integrand is kx^2 - x^3. What are the x-intercepts of this function? Those will be your limits of integration. Your integral is a definite integral, so the constant C is unnecessary.

Dang. That was what I was looking for...

The x intercepts of f(x) I found was x = 0, x = k.

I integrated f(x) with lower limit x = 0 and upper limit x = k .. and equaling this integration to 2.

I get a k value of 24^(1/4). Is this correct?

 

[Mark44]

Yep.