Spherical Integration Example, Multivariable Calculus

We rewrite a triple integral written in rectangular (xyz) coordinates in spherical coordinates. This is a multi-step process: first we sketch the domain of integration, then we figure out the bounds for the spherical coordinates, then we perform the integration. (This integral would be simpler in polar/cylindrical coordinates, but the point of the exercise is really describing the domain with spherical coordinates.) Recall the coordinates: - Radius (𝜌) Definition: In all examples, 𝜌 represents the radial distance from the origin to a point. - Angle 𝜙 Definition: 𝜙 is defined as the angle from the positive 𝑧-axis, analogous to latitude. - Angle 𝜃 Definition: 𝜃 represents the rotational angle in the 𝑥𝑦-plane, analogous to longitude. Our computation begins with a visual exploration of the domain of integration in the 𝑥𝑦-plane. This initial step is crucial, as it lays the foundation for our understanding of the problem in a three-dimensional context. We consider a triple integral whose limits are defined in rectangular coordinates, with a specific focus on the outer bounds, which represent a two-dimensional domain. The outer bounds suggest that the variable 𝑦 spans from −𝑅 to 𝑅, creating a region confined within two horizontal lines. Next, the middle bounds dictate that 𝑥 is confined between 0 and sqrt(𝑅^2−𝑦^2. This limitation of 𝑥 to non-negative values, coupled with the equation 𝑥^2+𝑦^2=𝑅^2, reveals that our domain is not an entire disk but rather the right semidisk of radius 𝑅 in the 𝑥𝑦-plane. Next, we visualize the 𝑥𝑦-plane within the larger context of 𝑥𝑦𝑧-space. Here, the third variable, 𝑧, comes into play, ranging from the surface described by 𝑧=sqrt(𝑥2+𝑦2) up to the plane 𝑧=𝑅. In essence, the domain we are dealing with is a solid half cone, originating from the right semicircle in the 𝑥𝑦-plane and extending upward to the plane 𝑧=𝑅. Our task is to redefine the domain of integration in these spherical terms. Theta (𝜃) Bounds: The domain extends from the negative to the positive y-axis, thus 𝜃 ranges from −𝜋/2 to 𝜋/2. Phi (𝜙) Bounds: The angle 𝜙 starts from 0 (along the positive 𝑧-axis) and opens down to the edge of the cone, which is determined to be 𝜋/4. Radius (𝜌) Bounds: The radial distance 𝜌 begins at 0 (at the origin) and extends to a maximum determined by the edge of the cone. This maximum distance varies with 𝜙, and is calculated as 𝑅/cos(𝜙). Then we proceed in the video with the integration! Multivariable Calculus Unit 5 Lecture 5, supplement. #mathematics #multivariablecalculus #SphericalCoordinates #mathematicstutorials #tripleintegral #tripleintegration #matheducation #physicseducation #polarcoordinates #engineeringmath #iitjammathematics #calculus3