# How can I find the value of these types of integrals?

Let $R=\{(x,y,z): -2\leqslant z\leqslant xy\;\mathrm{ and }\;x^2+y^2=1\}\subset\mathbf R^3$ and consider the vector field $\mathbf F(x,y,z)=-x\mathbf i+y\mathbf j+\exp(z^2)\mathbf k.$ I want to find $$\iint_R\mathrm{curl}\;\mathbf F\;\mathrm d S,$$ but I do not know how to choose a suitable parametrisation.

How do you solve these type of integrals is general?

-

## 1 Answer

Hint: did you try calculating the curl of $\bf F$?

-

2
I found curl $\mathbf F=0$, which means that the integral is zero. – zeke Jul 23 '12 at 22:56
{"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": 4, "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}, "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": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9136472344398499, "perplexity_flag": "head"}