SUNY Geneseo Department of Mathematics
Monday, April 17
Math 223
Spring 2023
Prof. Doug Baldwin
(No.)
…will be Thursday, May 18 (the last day of finals).
There will probably be 2 more problems sets, although with slightly shorter completion and grading windows than until now, so that grading for both can at least nominally fit in before the end of classes.
Base on “Polar Rectangular Regions of Integration” in section 4.3 of the textbook.
You can do integration in polar coordinates, and it’s useful for integrals involving circular regions.
For polar integrals, the abstract differential area, “dA,” becomes “r dr dΘ” in iterated integrals.
Remember the conversion from rectangular to polar coordinates: x = r cosΘ, y = r sinΘ.
An example that involves conversion from rectangular to polar coordinates: Geometry says that the volume of a hemisphere of radius 1 is 2π/3. Do you get the same answer if you integrate the hemisphere z = √(1 - x2 - y2) over a circle of radius 1 around the origin?
Integrating the equation for the hemisphere in rectangular form would involve lots square roots, both from the equation for the hemisphere itself and from the bounds of the integrals. Writing the equation for the hemisphere in polar form and then integrating in polar coordinates is easier. In fact, the polar form of the hemisphere is even simpler than the rectangular form:
Integrating this starts with figuring out the bounds: 0 to 2π for Θ, because the hemisphere covers a full circle, and 0 to 1 for r, since points anywhere from the origin to 1 unit away from it can lie in the hemisphere.
Use a u-substitution to do the integration…
Reassuringly, the integral did give the same result as geometry said it should.
An example that’s in polar form from the beginning: The State University of New West Dakota is building a new library that will be a tower of radius 10 meters. Around half of the base of this tower will be a glassed-in seating/study area, which projects 4 meters beyond the edge of the tower, extending up until it meets a slanted roof defined by the equation z = 20 - r where r is radius measured from the center of the tower. What is the total volume of this part of the library?
The height of the seating/study area is the 20 - r height of its roof, so integrate this expression over the floor space of the area, namely over any π-radian range of angles (height doesn’t depend on the actual angle) and from r = 10 (the inner edge of the area) to 14 (the outer edge):
There’s one way to integrate a multivariable function that we haven’t thought about yet: along a curve in the space of the function’s variables. In 2 dimensions, this gives the area of a sheet between the curve and the graph of the function. The integral is called a “scalar line integral.”
Please read “Scalar Line Integrals” in section 5.2 of the textbook.