15.1: Double Integrals over General
Regions
Anthony Rizzie
University of Connecticut
Fall 2016
learning objectives
After today, you should be able to:
1. Understand how to set up a general double integral
2. Know how to switch the order of integration
3.
16.2: Line Integrals
(Scalar Integrals)
Anthony Rizzie
University of Connecticut
Fall 2016
learning objectives
After today, you should be able to:
1. Picture what a line integral of a scalar function means
2. Compute a line integral of a scalar function
3
warm-up question
~0 for the function
f px, y q x2 4x 2y 2 3?
~
At what point(s) is f
x2x 4, 4yy x0, 0y fx 2x 4 0, fy 4y 0.
Therefore, we have 2x 4 x 2 and 4y 0 y 0.
~
f
The only point where both of these conditions is satisfied is
p2, 0q.
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14.7: M
warm-up question
Assume that p2, 1q is a critical point for some function f px, y q.
If fxx p2, 1q 3, fxy p2, 1q 3, and fyy p2, 1q 4, then what
type of point is p2, 1q? Explain.
We dont know the function f px, y q, but we can compute the
value of Dpx, y q
15.8: Spherical Coordinates
Anthony Rizzie
University of Connecticut
Fall 2016
learning objectives
After today, you should be able to:
1. Understand and use spherical coordinates to describe a
region E
2. Decide which coordinate system is best for a given
warm-up question
Evaluate the iterated integral
54
2
54
2
1
3 dy dx 3
54
2
1
1
3 dy dx.
1 dy dx 3ApRq 3 3 5 45
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15.1: Double Integrals over General
Regions
Anthony Rizzie
University of Connecticut
Fall 2016
learning objectives
After today, you shoul
15.1: Double Integrals over
Rectangles
Anthony Rizzie
University of Connecticut
Fall 2016
learning objectives
After today, you should be able to:
1. Understand the concept of a double integral
2. Know when a double integral is computing volume or not
and
warm-up question (ex. 4 continued)
Sketch the region E that corresponds to the integral
1 1 1y
?x 0 f px, y, z q dz dy dx
0
as well as the shadows in each of the coordinate planes.
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example 4
Now for the tricky part.use the shadows to write the oth