<center><a href='https://dtube.video/#!/v/masterwu/6dfqg3vt'><img src='https://gateway.ipfs.io/ipfs/QmUSdWUM1VeG2MvBdWRWGjH9W92JH8AcndCU7JBynQPE59'></a></center><hr>What I love about mathematics is some of the very creative and ingenious methods that someone has come up with to find solutions to perplexing problems. <center></center> The integral of √(<em>a</em><sup>2</sup> - <em>x</em><sup>2</sup>) is a great example one of these problems. If you look up a table of integrals, you may find the solution to look something like this... <center></center> Well, you might expect √(<em>a</em><sup>2</sup> - <em>x</em><sup>2</sup>) to be in the answer, but how did the term arcsin(<em>x</em>/<em>a</em>) get in there? How did we start with a purely algebraic expression, and end up with one having a trigonometric function? You might have also guessed we needed to make a substitution in order integrate this function. So the answer lies in the approach to the substitution. If we simply made the substitution... <center> </center> ... we would quickly run into problems. We need another approach. If you consider √(<em>a</em><sup>2</sup> - <em>x</em><sup>2</sup>) as an expression for solving a Pythagorean Theorem problem, we can set up a right-angle triangle where <em>a</em> is the hypotenuse, and <em>x</em> as the length of the vertical side, as depicted above. From trigonometry, sin<em>θ</em> = <em>x</em>/<em>a</em>. So, what if we made the substitution <em>x</em> = <em>a</em>sin<em>θ</em>? We get... <center></center> Geometrically, <em>a</em> must be positive, but it doesn't have to be numerically. This is not of any significant consequence since the squaring of <em>a</em> turns it positive everywhere else. Let's just assume that <em>a</em> is positive, so we can drop the absolute value bars. Let's also assume that... <center></center> ...which means that... <center></center> ...and thus we can drop the absolute value bars on the cos<em>θ</em> term. Thus the integrand becomes... <center></center> Now, we need a substitution for <em>dx</em> as well. So we simply take the derivative of <em>x</em> = <em>a</em>sin<em>θ</em>. <center></center> So, the integral becomes... <center></center> Now, to integrate cos<sup>2</sup><em>θ</em>, we need the half-angle formula... <center></center> Substituting this in, we get... <center></center> Now, another trigonometric identity is... <center></center> Putting this in, we get... <center></center> Before we finish, we need to get everything back in terms of <em>x</em>'s, rather than leave the solution in terms of <em>θ</em>'s. We established that... <center></center> Also from trigonometry... <center></center> Then finally... <center></center> Wow! That was quite a process. If you need a clearer explanation, play my video. Thanks for watching. Please give me an <strong>Upvote</strong> and <strong>Resteem</strong> if you have found this video helpful. Please ask me a math question by commenting below and I will try to help you in future videos. I would really appreciate any small donation which will help me to help more math students of the world. Tip me some DogeCoin: A4f3URZSWDoJCkWhVttbR3RjGHRSuLpaP3 Tip me at PayPal: https://paypal.me/MasterWu<hr><a href='https://dtube.video/#!/v/masterwu/6dfqg3vt'>► Watch on DTube</a><br /><a href='https://gateway.ipfsstore.it:8443/ipfs/QmdpWf4ynmqZDLdZef5wcULiVCGFPQjggWMbTvJ5hYBA4r'>► Watch Source (IPFS)</a>
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| json_metadata | "{"format": "markdown", "links": ["https://dtube.video/#!/v/masterwu/6dfqg3vt", "https://paypal.me/MasterWu", "https://gateway.ipfsstore.it:8443/ipfs/QmdpWf4ynmqZDLdZef5wcULiVCGFPQjggWMbTvJ5hYBA4r"], "app": "steemit/0.1", "tags": ["dtube", "dtube-steemiteducation", "dtube-steemstem", "dtube-mathematics", "dtube-calculus"], "video": {"content": {"videohash": "QmdpWf4ynmqZDLdZef5wcULiVCGFPQjggWMbTvJ5hYBA4r", "tags": ["dtube", "dtube-steemiteducation", "dtube-steemstem", "dtube-mathematics", "dtube-calculus"], "description": "What I love about mathematics is some of the very creative and ingenious methods that someone has come up with to find solutions to perplexing problems.\n\n<center></center>\n\nThe integral of \u221a(<em>a</em><sup>2</sup> - <em>x</em><sup>2</sup>) is a great example one of these problems. If you look up a table of integrals, you may find the solution to look something like this...\n\n<center></center>\n\nWell, you might expect \u221a(<em>a</em><sup>2</sup> - <em>x</em><sup>2</sup>) to be in the answer, but how did the term arcsin(<em>x</em>/<em>a</em>) get in there? How did we start with a purely algebraic expression, and end up with one having a trigonometric function?\n\nYou might have also guessed we needed to make a substitution in order integrate this function. So the answer lies in the approach to the substitution. If we simply made the substitution...\n\n<center> </center>\n\n... we would quickly run into problems. We need another approach.\n\nIf you consider \u221a(<em>a</em><sup>2</sup> - <em>x</em><sup>2</sup>) as an expression for solving a Pythagorean Theorem problem, we can set up a right-angle triangle where <em>a</em> is the hypotenuse, and <em>x</em> as the length of the vertical side, as depicted above.\n\nFrom trigonometry, sin<em>\u03b8</em> = <em>x</em>/<em>a</em>. So, what if we made the substitution <em>x</em> = <em>a</em>sin<em>\u03b8</em>? We get...\n\n<center></center>\n\nGeometrically, <em>a</em> must be positive, but it doesn't have to be numerically. This is not of any significant consequence since the squaring of <em>a</em> turns it positive everywhere else. Let's just assume that <em>a</em> is positive, so we can drop the absolute value bars.\n\nLet's also assume that...\n\n<center></center>\n\n...which means that...\n\n<center></center>\n\n...and thus we can drop the absolute value bars on the cos<em>\u03b8</em> term. Thus the integrand becomes...\n\n<center></center>\n\nNow, we need a substitution for <em>dx</em> as well. So we simply take the derivative of <em>x</em> = <em>a</em>sin<em>\u03b8</em>.\n\n<center></center>\n\nSo, the integral becomes...\n\n<center></center>\n\nNow, to integrate cos<sup>2</sup><em>\u03b8</em>, we need the half-angle formula...\n\n<center></center>\n\nSubstituting this in, we get...\n\n<center></center>\n\nNow, another trigonometric identity is...\n\n<center></center>\n\nPutting this in, we get...\n\n<center></center>\n\nBefore we finish, we need to get everything back in terms of <em>x</em>'s, rather than leave the solution in terms of <em>\u03b8</em>'s. We established that...\n\n<center></center>\n\nAlso from trigonometry...\n\n<center></center>\n\nThen finally...\n\n<center></center>\n\nWow! That was quite a process. If you need a clearer explanation, play my video.\n\nThanks for watching. Please give me an <strong>Upvote</strong> and <strong>Resteem</strong> if you have found this video helpful.\n\nPlease ask me a math question by commenting below and I will try to help you in future videos.\n\nI would really appreciate any small donation which will help me to help more math students of the world.\n\nTip me some DogeCoin: A4f3URZSWDoJCkWhVttbR3RjGHRSuLpaP3\nTip me at PayPal: https://paypal.me/MasterWu"}, "info": {"snaphash": "QmUSdWUM1VeG2MvBdWRWGjH9W92JH8AcndCU7JBynQPE59", "author": "masterwu", "title": "Integral of \u222b\u221a(a^2 - x^2)dx using a Trigonometric Substitution", "permlink": "6dfqg3vt"}, "_id": "7fa07229b79881f6b95800bf3d4623ab"}, "image": ["https://gateway.ipfs.io/ipfs/QmUSdWUM1VeG2MvBdWRWGjH9W92JH8AcndCU7JBynQPE59"]}" | ||||||
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Very nice completion of post! @masterwu
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Thank you @gamzeuzun
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