Solving Volume Problem with f(x)=kx2+x3

[nns91]

Homework Statement

f(x)=kx²+x³, where is k is a positive constant. Let R be the region in the first quadrant bounded by the graph of F and the x-axis.

a. Find all values of the constant k for which the area of R equals 2.
b. For k>0, write an integral expression in terms of k for the volume of the solid generated when R is rotated around x-axis.
c. For k>0, write an expression in terms of k, involving one or more integrals, that gives the perimeter of R.

Homework Equations

V= pi *r²

The Attempt at a Solution

a. For a. I find the interception of F(x) and the x-axis. After that, I set up an integral of f(x) evaluating from 0 to k and set that integral equal to 2. Am I right.

b. and c. I don't really get the part of in terms of k. so I just do a normal integration of dish method of f(x) and I substitute the number in part a for k. Am I right ? Or is it trickier ?

 

[Mark44]

Are you sure you have the right formula for f? The formula that you gave, f(x) = kx^2 + x^3, has an intercept at (0, 0) and no others on the positive x-axis. This means that the area in the first quadrant between the graph of f and the x-axis is infinite.

In you attempt to find the answer to part a, why would you have k as a limit of integration? Before going on to parts b and c, make sure that you are giving us the correct information for this problem.

 

[nns91]

I am sorry it's minus not plus.

Because k is a x-intercept ?

Am I right ?

 

[HallsofIvy]

What's "minus not plus"!

Do you mean it should be y= kx²- x³? In that case y= x²(k- x) so x= 0 is a double root and x= k is a root.

If you rotate the curve, for 0< x< k, around the x-axis each cross section perpendicular to the x-axis is a circle with radius y= kx²- x³. The area of such a circle is $k\pi y^2$.

The perimeter of R consists of two parts: the curve between x= 0 and x= k and the straight line from (0,0) to (k, 0).

 

[nns91]

Can you explain about part b ? Why there is k there ? I got only pi*^2

 

[Mark44]

In part b, the volume of a typical volume element is
$$\Delta V = \pi y^2 \Delta x$$
and not $$k \pi y^2$$ as Halls said.

or,
$$\Delta V = \pi (x^2(k - x))^2 \Delta x$$

So what should the integral look like? (All you need to do is write the integral.)

I took the problem further and calculated the integral, getting:
$$\pi k^7 [\frac{1}{5} - \frac{1}{3} + \frac{1}{7}]$$

 

[HallsofIvy]

Can you explain about part b ? Why there is k there ? I got only pi*^2

I have no idea where the k came from! It snuck in while I wasn't looking!

 

[nns91]

what do you guys think the " in term of k" means ? Do I just write a normal integral or is it trickier ?

 

[Mark44]

"In terms of k" means that when you write the integral expressions, they will involve k, which is unspecified. For parts b and c, you don't actually have to evaluate the integrals--at least that's how I interpret the instructions.

 

[nns91]

So do I substitute the k I found in part a in or just leave it as k ?