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Double Integrals in Cylindrical Coordinates: Calculating Areas and Volumes, Study notes of Calculus

Portland Community College Calculus

Instructions on how to calculate the areas and volumes of various shapes using double integrals in cylindrical coordinates. It includes examples of finding the areas of regions inside and outside of circles and limacons, as well as the volume of solids bounded by paraboloids. The document also includes conversions between polar and rectangular coordinates.

Typology: Study notes

Pre 2010

Uploaded on 08/19/2009

koofers-user-uzl 🇺🇸

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Double Integrals using Cylindrical Coordinates

Page 1 of 8

Plot the polar function

()

11.5sinr

Graph on your graphing utility:

2cos2

⎛⎞

=+ ⎜⎟

⎝⎠

()

cos 3

2cos 4 sin 4

⎛⎞

=− +

⎜⎟

⎝⎠

Table 1:

()

11.5sinr

Discover Study notes of Calculus Portland Community College

Partial preview of the text

Download Double Integrals in Cylindrical Coordinates: Calculating Areas and Volumes and more Study notes Calculus in PDF only on Docsity!

Plot the polar function r = 1 + 1.5sin( θ).

Graph on your graphing utility:

2 cos 2

( )

cos 3 2cos 4 sin 4

r e

Table 1: r = 1 +1.5sin( θ)

θ r

Example 1

Find the area of the region inside the circle r = 3sin( θ) and outside the convex limacon

r = 2 − sin ( θ ).

Fact

dA ∫∫

finds the (numerical) area of the region R.

Double Integrals in Cylindrical Coordinates ( r , θ, z )

A double integral in cylindrical coordinates always has the form

stop out

start in

f r r dr d

∫ ∫

where rin = fin ( θ)is the boundary curve closest to the pole, rout = fout ( θ)is the boundary curve

farthest from the pole, the region of integration is swept out (exactly once) in the counterclockwise

direction from θ = θ start to θ = θ stop , and dA = r dr d θ.

Note: If the region of integration includes the pole, then rin = 0.

Example 4

Find the area of the region common to the interiors of the dimpled limacons r = 1.5 − sin( θ )and

r = 1.5 − cos ( θ).

Example 5

Find the volume of the solid bounded by the paraboloids

2 2 z = 4 − x − y and

2 2 z = 3 x + 3 y.

Polar to Rectangular Rectangular to Polar

x = cos ( θ) y = sin( θ)

2 2 2 r = x + y

2 2 2

x + y = r tan ( )

Convert each function to polar form.

x = 7 y = 3

2 2 x + y = y = x

2 2

x + y + x = ( )

2 2 2 2 x x + y = y

−

2 2

cos x y dxdy

∫ ∫

4 −

x^2 dydx