## Calculus (3rd Edition)

$\frac{8}{27}((1+(9/2)\pi)^{3/2}-1)$
We have $r(t) =\lt \cos t,\sin t, t^{3/2}\gt$, and so the vector derivative is $r'(t) =\lt -\sin t,\cos t, \frac{3}{2} t^{1/2}\gt$. We calculate the length by integration: $$length=\int_{0}^{2\pi}\sqrt{\sin^2 t+\cos^2t+(9/4)t} \ dt=\int_{0}^{2\pi}\sqrt{1+(9/4)t} \ dt=\frac{4}{9(3/2)}(1+(9/4)t)^{3/2}|_{0}^{2\pi}=\frac{8}{27}((1+(9/2)\pi)^{3/2}-1).$$
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "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": 0.99876868724823, "perplexity": 896.1792145882931}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "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-2020-05/segments/1579250615407.46/warc/CC-MAIN-20200124040939-20200124065939-00321.warc.gz"}