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semesters >
fall 2024 >
mth261 >
week 9
MTH-261 Week 9 (October 20-26)
semesters >
fall 2024 >
mth261 >
week 9 >
outline
15.4 Double Integrals in Polar Regions
recommended exercises
5, 9, 11, 13, 21, 25, 31, 34, 37, 39
learning objectives
recognize the format of a double integral over a polar rectangular region
evaluate a double integral in polar coordinates by using an iterated integral
recognize the format of a double integral over a general polar region
use double integrals in polar coordinates to calculate areas and volumes
additional notes
reasons for converting to polar coordinates
to make the limits simpler
to make the integrand simpler
converting integrands and limits of integration
the price of conversion
the conversion factor forshadows other conversions and other conversion factors
15.5 Applications of Double Integrals
recommended exercises
5, 9, 13, 17, 24, 27, 29, 32
learning objectives
illustrate various applications of double integrals
additional notes
density and mass
moments and centers of mass
moments of inertia
about the axes
about the origin
radii of gyration
Probability
joint density functions, independence, expected values
15.6 Surface Area
recommended exercises
4, 6, 7, 11, 13, 17, 19, 23
learning objectives
find the parametric representations of a cylinder, a cone, and a sphere
describe the surface integral of a scalar-valued function over a parametric surface
use a surface integral to calculate the area of a given surface
explain the meaning of an oriented surface, giving an example
describe the surface integral of a vector field
use surface integrals to solve applied problems
additional notes
the area of a surface \( S: \vec{r}(u,v) \)
\( \displaystyle{\int\int_{S}\,dS = \int\int_{R} \left\|\frac{\partial \vec{r}}{\partial u}\times\frac{\partial \vec{r}}{\partial v}\right\| \,dA }\)
the contribution of a function to a surface \( S: \vec{r}(u,v) \)
\( \displaystyle{\int\int_{S}f\,dS = \int\int_{R} f\left(\vec{r}(u,v)\right)\left\|\frac{\partial \vec{r}}{\partial u}\times\frac{\partial \vec{r}}{\partial v}\right\| \,dA }\)
orientable surfaces and flux
\( \displaystyle{\int\int_{S}\vec{F}\cdot\vec{N}\,dS = \int\int_{R} \vec{F}\left(\vec{r}(u,v)\right)\cdot\left(\frac{\partial \vec{r}}{\partial u}\times\frac{\partial \vec{r}}{\partial v}\right) \,dA }\)
semesters >
fall 2024 >
mth261 >
week 9 >
assessments
Assessments
Deadlines
All homework is due at the beginning of class on the date indicated.
All exams are given in class on the date indicated.
Late homework is not accepted and there are no make-ups for missed exams.
Contact me before the deadline if you have a conflict with a
deadline and we will find a mutually agreeable solution.
Solutions are posted under these links after the deadline.
Homework
due Thursday, October 24:
due Thursday, October 31:
due Tuesday, November 5:
Exams
Thursday, November 7: Exam 2
Exam 2
(solutions)
Exam 2 covers sections 14.4-14.8 and 15.1-15.7
semesters >
fall 2024 >
mth261 >
week 9 >
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Formula Sheets
Additional Exam Practice
Cylindrical and Spherical Coordinates
Calculus Facts
Multivariable Functions
Green's, Gauss', Stokes' Theorems
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fall 2024 >
mth261 >
week 9 >
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