SUNY Geneseo Department of Mathematics

Integration in Polar Coordinates, Part 2

Tuesday, November 8

Math 223 01
Fall 2022
Prof. Doug Baldwin

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Misc

PRISM Group Study Session

This Sunday, November 13, 2:00 - 5:00 PM in South 340.

A chance to work on math with like-minded people to ask questions of, help out, or just socialize with.

Drop in and leave whenever you want.

Colloquium

On mathematical modeling of biomedical phenomena (specifically “Postural Orthostatic Tachycardia Syndrome”).

By Justin Geddes, 2018 Geneseo graduate and current Ph. D. student at North Carolina State University.

Justin will also talk about summer undergraduate research (REU) opportunities in biomathematics at NCSU.

Friday, 3:30 - 4:30, Newton 204.

Integration in Polar Coordinates

Finishing the cylinder problem from yesterday.

What is the volume inside the cylinder x2 + y2 = 1 and between the planes z = y + 2 and z = -(x+2) = -x - 2?

Vertical cylinder with ends cut by slanted planes, planes slant in directions perpendicular to each other

The first step is to express the planes’ equations in polar form (you can also put the cylinder in polar form, but that turns out to be less helpful in the long run). Then the volume you want is the integral of the cylinder’s length, over all radius values between 0 and 1, and all angles between 0 and 2π:

Finding volume of cylinder with ends cut at slants by integrating distance between ends

Once you’ve set up the integral, evaluate it from innermost integral to outer, as you would other iterated integrals:

Integrating R sine Theta plus R cosine Theta plus 4 all times R from 0 to 1 and 0 to 2 Pi yields 4 Pi

Next

Mass and center of mass problems as integration.

Some background information:

Informally, “mass” is weight.

“Density” is mass per unit volume.

One consequence of these definitions is that if an object has a density that varies with position, ρ(x,y,z), you can find its total mass by integrating that density over the object:

Blob V with density rho of X, Y, Z; mass is integral over V of rho

Please read “Center of Mass in Two Dimensions” in section 4.6, and enough of “Center of Mass and Moments of Inertia in Three Dimensions” to appreciate how the ideas in the first section extend to 3 dimensions.

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