Triple Integrals Between Spheres in Spherical Coordinates | Calculus 3 Tutorial ( Step by Step )
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In this Calculus 3 tutorial, we demonstrate how to evaluate a triple integral using spherical coordinates. This method is perfect for handling multivariable calculus problems where the region of integration lies between two spheres. By using spherical coordinates, we can simplify the integrand and make the calculation much easier.
We begin by identifying the region EEE bounded by two spheres, with the spherical radius ρ\rhoρ ranging from 2 to 6. This helps us set up the proper bounds for the triple integral.
In this video, we’ll guide you through:
Separating the integrals: Breaking down the triple integral by recognizing the independence of the variables.
Evaluating each integral: Simplifying the integrals one by one.
Combining the results: Bringing the evaluated integrals together to find the final solution.
Watch this tutorial to understand how to apply spherical coordinates for solving triple integrals in multivariable calculus!
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
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