How are triple integrals defined in cylindrical and spherical coor-dinates? Why might one prefer working...

boitshupoO

boitshupoO

Answered

2020-12-28

How are triple integrals defined in cylindrical and spherical coor-dinates? Why might one prefer working in one of these coordinate systems to working in rectangular coordinates?

Answer & Explanation

Nicole Conner

Nicole Conner

Expert

2020-12-29Added 97 answers

Step 1 Give the notes about how the triple integrals defined in cylindrical and spherical coordinates. Step 2 The cylindrical coordinates denotes a point P in space by ordered triples (r,θ,z)tr^andθ are polar coordinates for the vertical projection of P on the xy-plane with rθ and z is the rectangular vertical coordinate. The equations related to the rectangular coordinates (x, y, z) and cylindrical coordinates (r,θ,z) are, x=rcosθ,y=rsinθ,z=z,r2=x2+y2 and tanθ=yx Step 3 The spherical coordinates represent a point P in space by ordered triples (p,ϕ,θ) in which, p is the distance from P to the origin (p0) ϕ is angle over{OP} makes with the positive z-axis (0ϕπ) θ is the angle from cylindrical coordinates. Step 4 The equations relating spherical coordinates to Cartesian and ctlindrical coordinates are, r=psinϕ
x=rcosθ=psinϕcosθ
z=pcosϕ
y=rsinθ=psinϕsinθ
p=x2+y2+z2=r2+z2
tanθ=yx Cylindrical coordinates are good for describing cylinders whose axes run along the z-axis and planes that either contain the z-axis or lie perpendicular to the z-axis. Surfaces like these have equations of constant constant coordinate value.

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