# What is the arc length of f(x) = x-xe^(x^2)  on x in [ 2,4] ?

Jan 1, 2018

${\int}_{2}^{4} \sqrt{1 + {\left(1 - {e}^{{x}^{2}} - 2 {x}^{2} {e}^{{x}^{2}}\right)}^{2}} \setminus \mathrm{dx} \approx 2.93$

#### Explanation:

The formula for arc length of $f \left(x\right)$ on the interval $\left[a , b\right]$ is:
${\int}_{a}^{b} \sqrt{1 - {\left(f ' \left(x\right)\right)}^{2}} \setminus \mathrm{dx}$

First we'll work out the derivative of our function:
$\frac{d}{\mathrm{dx}} \left(x - x {e}^{{x}^{2}}\right) = 1 - \frac{d}{\mathrm{dx}} \left(x {e}^{{x}^{2}}\right)$

To figure out the second part, we'll use the product rule:
$\frac{d}{\mathrm{dx}} \left(f \left(x\right) g \left(x\right)\right) = f ' \left(x\right) g \left(x\right) + f \left(x\right) g ' \left(x\right)$

In our case, we get:
$1 - \left(1 \cdot {e}^{{x}^{2}} + x \cdot \frac{d}{\mathrm{dx}} \left({e}^{{x}^{2}}\right)\right)$

We can use the chain rule, which says:
$f \left(g \left(x\right)\right) = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

This gives us:
$1 - \left({e}^{{x}^{2}} + x \cdot 2 x {e}^{{x}^{2}}\right)$

$1 - {e}^{{x}^{2}} - 2 {x}^{2} {e}^{{x}^{2}}$

If we then plug this into our formula, we get:
${\int}_{2}^{4} \sqrt{1 + {\left(1 - {e}^{{x}^{2}} - 2 {x}^{2} {e}^{{x}^{2}}\right)}^{2}} \setminus \mathrm{dx}$

This integral doesn't have an elementary answer that I've been able to find, but the value can be approximated to be roughly $2.93$
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 12, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://socratic.org/questions/what-is-the-arc-length-of-f-x-x-xe-x-2-on-x-in-2-4", "fetch_time": 1720840830000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-30/segments/1720763514484.89/warc/CC-MAIN-20240713020211-20240713050211-00820.warc.gz", "warc_record_offset": 432359460, "warc_record_length": 6383, "token_count": 605, "char_count": 1451, "score": 4.6875, "int_score": 5, "crawl": "CC-MAIN-2024-30", "snapshot_type": "latest", "language": "en", "language_score": 0.47286075353622437, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00009-of-00064.parquet"}