Solving a Puzzling Equation: The Search for an Answer

[regnar]

This problem does not make sense to me and plus I keep getting a different answer:
You have y=x^3 and x=y^3 these are rotated about x=-1. I put everything in terms of y and solved with respect to y. I set the problem like this:

$$\pi$$$$\int_0^1{(1 +{\sqrt[3]{y})^{2}} - (1 + y^{3})^{2}}dy$$

I foiled each function out:

$$\pi$$$$\int_0^1{(1+2{\sqrt[3]{y}+y^{2/3}) - (1+2y^{3}+y^{6})}dy$$
I'm not sure if I did this right.

Continued:

$$\pi$$$$\int_0^1{2{\sqrt[3]{y}+y^{2/3} - 2y^{3}-y^{6}}dy$$$$\pi$$$$[2(3/4)y^{4/3}+(3/5)y^{5/3}-(1/2)y^{4}-(1/7)y^{7}]^{1}_{0}$$$$\pi$$$$[(3/2) + (3/5) - (1/2) - (1/7)]- 0$$$$\pi$$$$[51/35]$$

 

[Mentallic]

Sorry, before I delve into the problem, does the question ask something like:
The area between the curves y=x³ and x=y³ for $0\leq x \leq 1$ is rotated about the line x=-1. Find the volume.

 

[Mark44]

Yes, please post the exact problem statement. Inquiring minds need to know.

 

[Mentallic]

regnar, everything that you've done is correct if the question asked for the section in $0\leq x\leq 1$. But if it didn't all you have to do is change the limits

 

[regnar]

Sorry, but how would I go about changing the limits and the question exactly what put on there it gave those two function and said it rotated about x=-1.

 

[Mentallic]

I meant the limits of integration - the limits a and b in $$\int_a^bf(x)dx$$

Well as you know, the graphs y=x³ and x=y³ have an infinite domain and range, so "rotate these graphs about x=-1" doesn't tell us much. We'll assume we only need to worry about the area between the graphs in the domain $0\leq x\leq 1$, but it could just as easily be asking for the domain of $-1\leq x \leq 0$ or anything for that fact. Your answer is right, but it depends on what the question intended.

 

[regnar]

I tried what you said and I got the same answer, but here is the exact question: "Find the volume of the solid obtained by rotating the region bounded by and about the line x = -1."

 

[Mark44]

I tried what you said and I got the same answer, but here is the exact question: "Find the volume of the solid obtained by rotating the region bounded by and about the line x = -1."

Bounded by what and what?

 

[regnar]

It doesn't specify but I'm guessing -1. I also set each function equal to each other and got 0 and 1.

 

[Mark44]

Aren't the equations y = x^3 and x = y^3 mentioned in the problem statement?

 

[regnar]

yes, I solved for x for y=x^3 and then set them equal to each other

 

[Mark44]

Please give the complete problem statement - that's what I'm asking.

 

[regnar]

Im sorry, but that is the exact statement our teacher has given us.

 

[Mark44]

I tried what you said and I got the same answer, but here is the exact question: "Find the volume of the solid obtained by rotating the region bounded by and about the line x = -1."

Is this it?
"Find the volume of the solid obtained by rotating the region bounded by y = x^3 and x = y^3 about the line x = -1."

 

[Mark44]

What do you have as the points of intersection of the two curves? There are three intersection points. I think you are working with only two of them.

 

[regnar]

I'm working with 0 and 1 but I also tried -1 and I got a negative answer unless I did that wrong which I'm not sure.

 

[Mark44]

You need two integrals for this problem - one for each of the two regions being rotated. The integral you have in post #1 is correct for the region in the first quadrant. For the region in the third quadrant, the two curves are in different places relative to each other. In other words, in the third quadrant, the outer radius is from x = y^3 and the inner radius is from x = y^(1/3).

BTW, the technique you are using is not disks - this is called the "washer" method. Each of your volume elements looks like a washer - a disk with a hole in the middle. (Not to be confused with a machine that washes clothes or dishes.)

 

[regnar]

Ohh, okay thank you very much. Also, thank you for clarifying the differences between disk and washer method.