Multivariable Calculus (Calc III) - Complete Semester Course - Compute a volume with a triple integral, three set ups

In this video, we work through the setup and computation of a triple integral that finds the volume enclosed between the plane z=2 and the sphere of radius 4 in the first octant of R^3. Our aim is not only to compute this volume, but also to understand how to describe the region accurately using inequalities in three different coordinate systems: rectangular, cylindrical, and spherical. We begin with a diagram to visualize the region and then proceed to set up the volume integral in rectangular coordinates. I will point out some common misconceptions students may have when working with bounds and emphasize the importance of understanding the geometric structure of the solid. We compute the volume explicitly in cylindrical coordinates via u-substitution and properties of iterated integrals. Finally, we describe the same solid using spherical coordinates, discussing how the angle φ and radial coordinate ρ vary throughout the region. Throughout the video, the focus remains on identifying the correct bounds for integration and understanding the structure of the volume element in each coordinate system. The computation illustrates why cylindrical coordinates are particularly effective in this case, even though all three approaches should yield the same result. #mathematics #math #multivariablecalculus #tripleintegral #SphericalCoordinates #calculus3 #mathlecture #integralcalculus #matheducation #STEMEducation #MathHelp #CoordinateSystems #FirstOctant