{"id":20231,"date":"2024-04-15T05:50:09","date_gmt":"2024-04-15T05:50:09","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20231"},"modified":"2024-04-15T05:50:09","modified_gmt":"2024-04-15T05:50:09","slug":"the-value-of-integral-mathopintintmkern-21mu-bigcirclimits_texts-overrightarrow-textr-cdot-overrightarrow-textn-textds-over-the-cl","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-value-of-integral-mathopintintmkern-21mu-bigcirclimits_texts-overrightarrow-textr-cdot-overrightarrow-textn-textds-over-the-cl\/","title":{"rendered":"The value of integral \\[\\mathop{{\\int\\!\\!\\!\\!\\!\\int}\\mkern-21mu \\bigcirc}\\limits_{\\text{S}} {\\overrightarrow {\\text{r}} \\cdot \\overrightarrow {\\text{n}} {\\text{ds}}} \\] over the closed surface S bounding a volume, where \\[\\overrightarrow {\\rm{r}} = {\\rm{x\\hat i}} + {\\rm{y\\hat j}} + {\\rm{z\\hat k}}\\] is the position vector and \\[{{\\rm{\\mathord{\\buildrel{\\lower3pt\\hbox{$\\scriptscriptstyle\\rightharpoonup$}} \\over n} }}}\\] is the normal to the surface S, is A. V B. 2V C. 3V D. 4V"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a99b553b72b8\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"true\">\r\n                    V                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    2V                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"false\">\r\n                    3V                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    4V                <\/div>\r\n                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    <!--more--><\/p>\n<p>The correct answer is $\\boxed{\\text{A) }V}$.<\/p>\n<p>The given integral is the surface integral of the dot product of the position vector and the surface normal. The surface normal is a vector <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> that is perpendicular to the surface, and its direction is determined by the right-hand rule. The dot product of two vectors is zero if they are perpendicular, so the surface integral will be zero if the surface normal is always perpendicular to the position vector. This is the case if the surface is a <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div> sphere, since the surface normal is always pointing directly away from the origin. Therefore, the value of the integral is the volume of the sphere, which is $V$.<\/p>\n<p>The other options are incorrect because they do not take into account the fact that the surface normal is always perpendicular to the position vector. If the surface is not a sphere, then the surface normal will not always be perpendicular to the position vector, and the value of the integral will be different from $V$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Join Our Telegram Channel Subscribe on YouTube<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[690],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The value of integral \\[\\mathop{{\\int\\!\\!\\!\\!\\!\\int}\\mkern-21mu \\bigcirc}\\limits_{\\text{S}} {\\overrightarrow {\\text{r}} \\cdot \\overrightarrow {\\text{n}} {\\text{ds}}} \\] over the closed surface S bounding a volume, where \\[\\overrightarrow {\\rm{r}} = {\\rm{x\\hat i}} + {\\rm{y\\hat j}} + {\\rm{z\\hat k}}\\] is the position vector and \\[{{\\rm{\\mathord{\\buildrel{\\lower3pt\\hbox{$\\scriptscriptstyle\\rightharpoonup$}} \\over n} }}}\\] is the normal to the surface S, is A. 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V B. 2V C. 3V D. 4V","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/the-value-of-integral-mathopintintmkern-21mu-bigcirclimits_texts-overrightarrow-textr-cdot-overrightarrow-textn-textds-over-the-cl\/","og_locale":"en_US","og_type":"article","og_title":"The value of integral \\[\\mathop{{\\int\\!\\!\\!\\!\\!\\int}\\mkern-21mu \\bigcirc}\\limits_{\\text{S}} {\\overrightarrow {\\text{r}} \\cdot \\overrightarrow {\\text{n}} {\\text{ds}}} \\] over the closed surface S bounding a volume, where \\[\\overrightarrow {\\rm{r}} = {\\rm{x\\hat i}} + {\\rm{y\\hat j}} + {\\rm{z\\hat k}}\\] is the position vector and \\[{{\\rm{\\mathord{\\buildrel{\\lower3pt\\hbox{$\\scriptscriptstyle\\rightharpoonup$}} \\over n} }}}\\] is the normal to the surface S, is A. 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