Parametric Calculus: Surface Area Part 2
Math Easy Solutions
In this video I go over Part 2 of the surface area for parametric curves video series, and this time continue on further from Part 1 and extend the surface formula to include the case where the parametric curve can’t be written in the form y = F(x). In other words, for the case where the parametric curve can’t be written as a one-to-one function. The derivation that I cover in this video is pretty extensive, but it nonetheless very similar to my earlier proof videos on the surface area for basic functions, as well like my arc length formula proof videos. The method goes back to what is known as Polygonal Approximation to first approximate the curve into straight line segments, then look at one line-segment and rotate that around the x-axis to obtain a surface, of which we can determine the surface area of it based on the circumference of a circle, and the length of the line segment. Once we determine that, we can then sum up the individual surface areas, and take the limit as the number of segments approaches infinity. Doing this, I show that what we get is, although not “exactly” a Riemann Sum, a summation that approaches to that of the definition of a Riemann Sum as the number of segments, or intervals approaches infinity. Thus putting this all together we get an integral, which is the exact same as in Part 1, thus extending its applicability to more complex types of shapes. This is an extensive proof video, but if you follow closely and pay attention, it will be an incredibly good learning experience to just some of the rigor that is involved in many of the more advanced mathematical proofs, so make sure to watch this video!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhuVcMj-AmgyrMlHYMw
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/parametric-calculus-surface-area-part-2
Related Videos:
Parametric Calculus: Surface Area Part 1: https://youtu.be/4bMEIf6WD8M Parametric Calculus: Arc Length Part 3 (Debunking My Own Video): https://youtu.be/udz4jzUgPBg Parametric Calculus: Arc Length Part 2: https://youtu.be/anD_j0nDDPA Parametric Calculus: Arc Length Part 1: https://youtu.be/AWvJDK-m6wQ Parametric Calculus: Areas: https://youtu.be/XdplYV61xlM Parametric Calculus: Tangents: https://youtu.be/deQwD2o0Sas Parametric Equations and Curves: https://youtu.be/Kd3XF4LZoFE Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk Simple Proof of the Pythagorean Theorem: http://youtu.be/yt-EJlbJQp8 Integration Overview: How are Riemann Sums, Antiderivatives, and Integrals Linked?: https://youtu.be/TGnnu1vnD_U Mean Value Theorem - A Simple Proof: http://youtu.be/x-2MiiG2E38 .
65135994 Bytes