Two arc length and one surface area problems

[about]

I been stuck on these for a while because I seemingly can't get integrals that can be solved. I got an error about temporary files or something when trying to attach images using links and the upload button wouldn't show up.

Find the arc length of of y = 1 - x^(2/3) from the point (-8,-3) to (-1,0).

Attempt:

,

2. Find the arc length of y = x^(4) + 1/(32x^2)

Attempt:

Find the area of the surcace obtained from rotating f(x) = 3(x)^(1/2), 0 =< x =< 2 around the x axis.

Attempt:

Now I know what you're thinking about the last one. "Hey, didn't you finish that one?" I know, it just can't be right precisely because I was seemingly able to finish it.

 

[tkhunny]

1) Your work is terribly messy. Please make an effort to be more organized and neat.
2) Please get those imgur links out of there. We don't control all that extra and inappropriate content. That's no good.
3) You have lots of scribbling. I couldn't find anywhere where you stated clearly your result.
4) Make sure you learn how to estimate. For the first, \(\displaystyle \sqrt{7^2 + 3^2} = \sqrt{58} = 7.616 < Actual\;Length\)

 

[mmm4444bot]

I got an error about temporary files or something when trying to attach images using links …

We can't upload an images to this server using a link; we need a file name, for that. Instead, use the Insert Image icon (found on the formatting menu bar). This allows importing a hosted-image from elsewhere, using its URL.

Also, please read the forum guidelines. Among other things, we prefer separate threads for separate exercises. :cool:

1. Find the arc length of of y = 1 - x^(2/3) from the point (-8,-3) to (-1,0)

You wrote:

y' = (2/3)·x^(-1/3)

There's a sign error. (This mistake did not affect your subsequent work because y' was squared.)


Later, you took the square root of 9x^(2/3) and got 3x^(7/6). That's not correct.

[9x^(2/3)]^(1/2)

9^(1/2) · [x^(2/3)]^(1/2)

3 · x^[(2/3)(1/2)]

Your mistake was adding the exponents, instead of multiplying them (as shown above).

2. Find the arc length of y = x^(4) + 1/(32x^2)

You typed 32 in the denominator, above, but your work shows it in the numerator.

You used 1+y' in the formula, instead of 1+[y']^2

I did not check any of the algebra because I'm not sure what y is.

3. Find the area of the surcace obtained from rotating f(x) = 3(x)^(1/2), 0 =< x =< 2 around the x axis.

Your setup looks good, but there are some algebra mistakes.

You squared (3/2)·x^(-1/2) and got (9/2)·x^0

When you squared 3/2, you forgot to square the denominator.

Also, you added the exponents. Maybe you were thinking:

x^(-1/2) · x^(-1/2) = x^(^-1/2 + ^-1/2)

If so, you added wrongly.

The other way to simplify a power of another power is like this:

[x^(-1/2)]^2 = x^[(-1/2)(2)]

That is, we multiply the exponents.