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Calculus 3, Triple Integral between two spheres in spherical coordinates

We show a triple integral best handled using spherical coordinates in this multivariable calculus problem. This method is ideal because the region of integration lies between two spheres, and the integrand simplifies significantly when expressed in spherical coordinates. To start, we identify the region E bounded by spheres where rho is the spherical radius, we can simplify our integrand and region of integration. Our task is to evaluate the triple integral. The radial bounds are defined by the spheres, the spherical radius ranging from 2 to 6. To simplify the evaluation: 1. Separate the integrals: Recognize that we can split the integral due to the independence of variables. 2. Evaluate each integral. 3. Combine the results. ------------------------------------ Chapters / Timestamps 00:00 Intro 00:38 Spherical Coordinates Spherical Geometry 00:58 Spherical Triple Integral Setup 02:11 Sketch of the Volume Region E 02:56 Spherical Coordinates bounds 06:19 Azimuthal Angle, Declination Angle, Spherical Geometry 10:04 Setup of Triple Integral 13:02 Seperate Integals as Product 16:31 Evaluation of Seperate Integrals 21:35 Wrap Up ------------------------------------ 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