Question

Calc III Tutorial: Integration in Cylindrical Coordinates

Answer 1

\({\bf{Cylindrical~Coordinates~Defined:}}\)
- consist of three components (r, θ, z) where
r = the distance from the origin to the point along the x-y axis
θ = the the angle from the positive x-axis to the ray containing r

r and θ work the same way they do in 2-d polar coordinates, but all we are doing is adding a third coordinate, z, to specify the location of the point along the z-axis



these are known as cylindrical coordinates because keeping r constant while varying θ and z creates a cylinder

Answer 2

\({\bf{Rectangular~to~Cylindrical:}}\) these work the same way as rectangular to polar in 2 dimensions but as a reminder

x = rcosθ
y = rsinθ
x^2 + y^2 = r^2
z = z
tanθ = y/x

Special cases
r = constant gives a cylinder with radius = the constant, and its axis along the z-axis
θ = constant, plane along θ that passes through the z-axis
z = constant, plane perp. to the z-axis at the constant

Answer 3

\({\bf{Cylindrical~Integrals:}}\)
\[\int\limits_{θ1}^{θ2}\int\limits_{r1}^{r2}\int\limits_{z1}^{z2}f~dz~r~dr~dθ\]

**important** you gotta include that r. if you just write dz dr dθ it's wrong

Method:

*** project the shape onto the xy axis. trust me this will save you a ton of headache when trying to visualize the limits on a 3d shape. basically you draw the shadow the image would cast if there was a light directly above the shape. ***

1. find the upper and lower limits along the z-axis. you can do this by drawing a straight vertical line through the shape and seeing the lower and upper intersections on the shape

2. find the upper and lower limits along r. you can do this by drawing a ray from the origin straight through the projection onto the xy axis and seeing the inner and outer intersections on the shape

3. find the θ limits of integration. this might be a lot easier if you **just** draw the x and y axis as normal and select the two theta values that enclose the projection exactly

Answer 4

from personal experience the theta limits are usually on the axis or on major unit circle values

Answer 5

Source material is section 15.7 of Thomas' Calculus, Early Transcendentals, 14th edition by Hass, Heil, Weir, et. al.