double integrals

Question 1

double integral meaning

Answer 1

given f(x,y) and a region D in the x-y plane the double integral is the signed volume of the region between z = f(x,y) and D in the z = 0 plane

Question 2

definition of the double integral (rectangular region)

Answer 2

for now let D be rectangular = [a0, a1] x [b0, b1]
and the subdivision S is constructed by specifying the points of a 2D lattice (corners of rectangles) with
a0 = x0 <x1..<xn = a1
b0 = y0 <y1..<yn = b1
the edge lengths of these are dxi = xi - xi-1 dyj = yj - yj-1
the norm |S| is the max length of the dxi and dyj
we take sample points (pi, qj) within [xi-1,xi] x [yj-1,yj] inside each rectangle
the area of the rectangle is dxidyj and we associate this to a cuboid of height f(pi,qj)
the volume is the Riemann sum
R = ∑ ∑ f(pi,qj) dxi dyj from j= 1 to n and i = 1 to n
and the double integral is the limit of R as |S| -> 0

Question 3

if f(x,y) is constant

Answer 3

f(x,y) = c
∫∫ f(x,y) dx dy = C * area(D)

Question 4

iterated integral

Answer 4

can be used to compute a double integral
valid if f is continuous
D = [a0,a1] x [b0,b1]
∫∫ f(x,y) dx dy = ∫( ∫ f(x,y) dy from b0 to b1) dx from a0 to a1
or can switch order of x and y
(fubinis theorem)

Question 5

y simple

Answer 5

if every line parallel to the y axis which intersects D does so in a single line segment or point

Question 6

double integral if y-simple

Answer 6

can be calculated as an iterated integral integrating over y first

Question 7

x simple

Answer 7

if every line parallel to the x axis which intersects D does so in a single line segment or point

Question 8

double integral if x-simple

Answer 8

can be calculated by an iterated integral integrating over x first

Question 9

double integral if y-simple and x-simple

Answer 9

the double integral can be calculated as an iterated integral in either order
might only be able to actually do the integral in one of the 2 orders

Question 10

double integral if neither y-simple or x-simple

Answer 10

if D is neither we may be able to split into a sum of regions which are x-simple or y-simple

Question 11

integration using polar coordinates

Answer 11

x = rcosθ
y = rsinθ
dA = dxdy = r dr dθ
if the region is r-simple or θ-simple we can do iterated integral as before

Question 12

jacobian

Answer 12

the determinant of the matrix of partial derivatives
J = d(x,y)/d(u,v) =
|dx/du dx/dv |
|dy/du dy/dv |

Question 13

area element in terms of jacobian

Answer 13

dA = dxdy = |J|dv du
where |J| is the absolute value of the jacobian

Question 14

gaussian integral

Answer 14

∫ e^(ax^2) from ∞ to -∞ for a >0
we derive this using a double integral
= √pi/√a