Calculus
Topics

# What is the arclength of f(x)=(1-x^(2/3))^(3/2)  in the interval [0,1]?

Jun 6, 2018

$\frac{3}{2}$

#### Explanation:

We get
$f ' \left(x\right) = - \frac{\sqrt{1 - {x}^{\frac{2}{3}}}}{x} ^ \left(\frac{1}{3}\right)$
and we have to solve
${\int}_{0}^{1} \sqrt{1 + \frac{1 - {x}^{\frac{2}{3}}}{x} ^ \left(\frac{2}{3}\right)} \mathrm{dx}$
simplifying the Integrand we get
${\int}_{0}^{1} {x}^{- \frac{1}{3}} \mathrm{dx} = {\left[\frac{3}{2} {x}^{\frac{2}{3}}\right]}_{0}^{1} = \frac{3}{2}$

##### Impact of this question
543 views around the world
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "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, "math_score": 0.8336068391799927, "perplexity": 12718.728709766005}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300574.19/warc/CC-MAIN-20220117151834-20220117181834-00305.warc.gz"}