Euler's Strawberry Daquari
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Math that you should care about.Mon, 11 Dec 2017 17:24:45 +0000enhourly1http://wordpress.com/https://secure.gravatar.com/blavatar/1316be8399c6abe9997ceb176b00c177?s=96&d=https%3A%2F%2Fs2.wp.com%2Fi%2Fbuttonw-com.pngEuler's Strawberry Daquari
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The integral of sin^2(x).
https://tcdty.wordpress.com/2012/07/29/the-integral-of-sin2x/
https://tcdty.wordpress.com/2012/07/29/the-integral-of-sin2x/#respondSun, 29 Jul 2012 18:08:14 +0000http://tcdty.wordpress.com/?p=8Continue reading →]]>I’m scheduled to teach Math 1320 this fall, so I’ve been writing some lecture notes. The course is simply the second part of the ‘standard’ calculus sequence, which means we’re going to be talking about ways to integrate just about anything before me move on to sequences and series, then Taylor’s theorem, and possibly some polar coordinate exercises. I’ll know more when I get the course memo. In any case, I’ve been going through the book and I’ve noticed that the following integrals are always done the same way:

and .

We make the substitution or , and then integrate that normally. There’s nothing wrong with this approach and students need to know it, but mathematicians are lazy and writing all that is too much work for such a simple integral. There’s a trick that you can do so that the integral becomes (almost) trivial.

Look at the graphs of and . Let’s say that for some . We know that

.

Thus we get that each of the above integrals is equal to the average of the two, i.e.

,

.

Since , we immediately see that the integral is equal to half of the length of the interval, namely

.

Sure, it only works for certain intervals, but it seems like whenever these problems come up has a ‘nice’ value that lets you do this.

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https://tcdty.wordpress.com/2012/07/29/the-integral-of-sin2x/feed/0spntddHello world!
https://tcdty.wordpress.com/2012/07/29/hello-world/
https://tcdty.wordpress.com/2012/07/29/hello-world/#commentsSun, 29 Jul 2012 15:51:17 +0000http://tcdty.wordpress.com/?p=1Continue reading →]]>Welcome to WordPress.com! This is your very first post. Click the Edit link to modify or delete it, or start a new post. If you like, use this post to tell readers why you started this blog and what you plan to do with it.