The spherical coordinates of a point P in 3-space are ρ (rho), θ, and ϕ (phi), where ρ is the distance from P to the origin, θ is the angle that t...The spherical coordinates of a point P in 3-space are ρ (rho), θ, and ϕ (phi), where ρ is the distance from P to the origin, θ is the angle that the projection of P onto the xy-plane makes with the positive x-axis, and ϕ is the angle between the positive z axis and the vector from the origin to P. When P has Cartesian coordinates (x,y,z), the spherical coordinates are given by
Given the difficulty of evaluating multiple integrals, the reader may be wondering if it is possible to simplify those integrals using a suitable substitution for the variables. The answer is yes, tho...Given the difficulty of evaluating multiple integrals, the reader may be wondering if it is possible to simplify those integrals using a suitable substitution for the variables. The answer is yes, though it is a bit more complicated than the substitution method which you learned in single-variable calculus.
This page covers the evaluation of triple integrals using cylindrical and spherical coordinates, emphasizing their application in symmetric regions. It explains conversions between coordinates, detail...This page covers the evaluation of triple integrals using cylindrical and spherical coordinates, emphasizing their application in symmetric regions. It explains conversions between coordinates, details the integration process, and showcases examples involving geometric shapes like cylinders, spheres, and paraboloids. The importance of Fubini's Theorem is highlighted for changing integration order. The text also includes exercises for practice and discusses calculating volumes of solid regions.
Sometimes, you may end up having to calculate the volume of shapes that have cylindrical, conical, or spherical shapes and rather than evaluating such triple integrals in Cartesian coordinates, you ca...Sometimes, you may end up having to calculate the volume of shapes that have cylindrical, conical, or spherical shapes and rather than evaluating such triple integrals in Cartesian coordinates, you can simplify the integrals by transforming the coordinates to cylindrical or spherical coordinates. For this topic, we will learn how to do such transformations then evaluate the triple integrals.
