Скачать с ютуб Calculus 3, Volume of a Cone using Cartesian, cylindrical, and also spherical coordinates в хорошем качестве

Calculus 3, Volume of a Cone using Cartesian, cylindrical, and also spherical coordinates

We find the volume of a cone using triple integrals in Cartesian, cylindrical, and spherical coordinates using multivariable calculus. Starting with Cartesian coordinates, we derive the limits of integration and set up the integral. The Z-coordinates range from 0 to 6/√3, while the X and Y-coordinates are bounded by the circle x² + y² = 36. Next, we convert the problem into cylindrical coordinates, noting that the radius r ranges from 0 to 6, and θ ranges from 0 to 2π. The z-coordinate limits are determined by the cone's height. Finally, we explore spherical coordinates, focusing on the region bounded by a constant declination angle, phi. We show how the transformation simplifies the volume calculation. Throughout the video, we emphasize the differences and benefits of each coordinate system for solving the integral. This video provides a comprehensive guide to using multiple coordinate systems to solve triple integrals for volume calculation. ------------------------------------ Chapters / Timestamps 00:00 Integral Setup 00:27 Height z bounds of cone 02:01 Horizontal x bounds, xy plane projectin 03:05 Intejection about right triangle angles 05:28 30-60-90 Right Triangle side ratios 07:23 Vertical y bounds, half circles, functions of x 09:30 Quadratic surfaces, circular cone 10:56 Solve for z(x,y) cone surface 15:36 Why integrand is 1. Other integrands discussion. 17:49 Cylindrical coordinates triple integral 19:32 Spherical coordinates triple integral 20:46 Find spherical coordinates radius in terms of phi 22:47 Take action! ------------------------------------ Please support us on Patreon: https://www.patreon.com/highpeakeduca... Please follow us on Facebook:   / high-peak-education-110762917338673   Please follow us on Twitter:   / highpeakeducate   Please follow us on Instagram:   / highpeakeducation   Please follow us on Reddit:   / highpeakeducation   Please follow us on TikTok:   / highpeakeducation   Interact with us on Twitch:   / highpeakeducation   Interact with us on Discord:   / discord   #HighPeakEducation #MultivariableCalculus #VectorCalculus