Integrals Example: Trigonometric Integral From Cardioid Video

in Threespeak2 months ago (edited)

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In this video I revisit my earlier video titled: Polar Coordinates: Arc Length: Example 1: Cardioid, but this time solve the resulting trigonometric integral manually. In that video I saved time in integrating the integral (2+2sinx)1/2 by instead using an online integral calculator. This is a very interesting integral and in solving it requires to thinking outside the box from the usual methodology of evaluating integrals. Because of the square root in the integrand, the integral can be better visualized if we can have it removed by manipulating the integrand, or as I showed in the video, by deriving our own trigonometric identity. I first off use the half-angle trigonometric identity because it involves both a squared cosine function (cos1/2(x)) and a cosine function (cos(x)). Thus I illustrate how we can go from what we have, the half-angle trig identity, to what we “want”, which is the function inside the square root. We can then realize that we have to apply cosine and sine identities to further simplify the function and eventually be able to solve the integral.

But in this video there is also a great lesson to learn, in that when dealing with squares we must always keep in mind that their value is always positive! Thus when simplifying the function to remove the square we must be aware if the resulting function must be taken as an absolute value. To remove the absolute values, we can separate the integral into intervals of positive and negative values of the integrand, and subtract the negative values to get a positive. From here I evaluate the integral obtaining the value of 8, which is the same as the arc length of the Cardioid shown in my earlier video.

This is an extremely important video because I utilize many integration techniques that I have hardly used before, so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvgSW4nC4kz_5VidWg

View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/video-notes-integrals-example-trigonometric-integral-from-cardioid-video

Related Videos:

Polar Coordinates: Arc Length: Example 1: Cardioid: https://youtu.be/PdZpkJR9d7s
Polar Coordinates: Arc Length Formula: https://youtu.be/lrh15Brq9B4
Polar Coordinates: Area Formula: Area Between Two Polar Curves: https://youtu.be/bAbz1T-_Twc
Polar Coordinates: Area Formula: https://youtu.be/wd-EwiKzqa0
Polar Coordinates: Graphing With Polar Curves with Desmos Calculator: https://youtu.be/xY9Be9HHRsk
Polar Coordinates: Cartesian Connection: https://youtu.be/HcaTYrpmGaU
Polar Coordinates: https://youtu.be/-KAdZL-N4ok
Parametric Equations and Polar Coordinates: https://youtu.be/usSors49Gdw
Half Angle Trigonometry Identities: http://youtu.be/0bY6tHZhBSI .


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