Handout 19 Recall: Given interval [a,b] we divide it into subintervals of equal size Δx , i.e. x j = a + jΔx, j = 0,1,...,n and define: 1) Reimann sum:
n
∑ f ( x ) Δx
f(x) *
i
i=1
2) Integration
b
∫
n
f ( x ) dx = lim ∑ f ( xi* ) Δx
n→∞
a
a i=1
b Double Integral For the rectangular domain [ a,b ] × [ c,d ] we divide the rectangle into small sub-­‐
rectangles Rij = [ xi−1 , xi ] × [ yi−1 , yi ] where xi = a + iΔx, j = 0,1,..., m, y j = c + jΔy, j = 0,1,...,n . The area of Rij is ΔA = Δx ⋅ Δy . The double Reimann sums which is also the approximation of the volume under surface z=f(x,y) is given by sums of volumes of the form (
)
m
(
n
)
f xij* , yij* ΔA , i.e. V ≈ ∑ ∑ f xij* , yij* ΔA . i=1 j=1
As usually we take the division of the rectangular into infinitely many infinitely small sub-­‐rectangles to get an exact value of the volume, thus V = lim
m,n→∞
∑ ∑ f ( x , y ) ΔA m
n
*
ij
*
ij
i=1 j=1
Definition: The double integral of z=f(x,y) over the rectangle R is given by ∫∫
f ( x, y ) dA = lim
R
m,n→∞
∑ ∑ f ( x , y ) ΔA if the limit exists. m
n
*
ij
*
ij
i=1 j=1
Conclusion: V = ∫∫ f ( x, y ) dA . R
Midpoint Rule: Let xi = a + iΔx, j = 0,1,..., m, y j = c + jΔy, j = 0,1,...,n and midpoints
xi =
m n
y + yj
xi−1 + xi
then ∫∫ f ( x, y ) dA ≈ ∑ ∑ f xi , y j ΔA . , y j = j−1
2
2
i=1 j=1
R
(
)
Average value: b
1
f ( x ) dx b − a ∫a
1
Given f(x,y) on rectangle R with area A(R) then fave =
f ( x, y ) ΔA A ( R ) ∫∫
R
Recall: Given f(x) on an interval [a,b] then fave =
Properties of Double Integrals: 1) ∫∫ f ( x, y ) + g ( x, y ) ΔA = ∫∫ f ( x, y ) ΔA + ∫∫ g ( x, y ) ΔA
R
R
R
2) ∫∫ cf ( x, y ) ΔA = c ∫∫ f ( x, y ) ΔA
R
R
3) f ( x, y ) ≥ g ( x, y ) ⇒ ∫∫ f ( x, y ) ΔA ≥ ∫∫ g ( x, y ) ΔA
R
R
Iterated Integrals: d
Suppose f ( x, y ) defined on rectangle R = [ a,b ] × [ c,d ] and let g ( x ) = ∫ f ( x, y ) dy where c
d
the expression ∫ f ( x, y ) dy is understood as a partial integration with respect to y, i.e. the c
variable x considered a constant for the process of integration. Next we integrate g to get b
b d
⎧
⎫
∫a g ( x ) dx = ∫a ⎨⎩⎪ ∫c f ( x, y ) dy ⎬⎭⎪ dx The last integral is called an iterated integral; we often omit the brackets and recognize 2 of them: b d
b
d b
d
⎧d
⎫
⎧b
⎫
1) ∫ ∫ f ( x, y ) dy dx = ∫ ⎨ ∫ f ( x, y ) dy ⎬ dx
2) ∫ ∫ f ( x, y ) dx dy = ∫ ⎨ ∫ f ( x, y ) dx ⎬ dy ⎪c
⎪a
a c
a ⎩
c a
c ⎩
⎭⎪
⎭⎪
Theorem (Fubini’s): If f(x,y) is continuous on rectangle R = [ a,b ] × [ c,d ] (or at least bounded with discontinuities on a finite number of smooth curves) then ∫∫
R
b d
d b
a c
c a
f ( x, y ) dA = ∫ ∫ f ( x, y ) dy dx = ∫ ∫ f ( x, y ) dy dx Thm: Let f(x,y) = g(x)h(y) then b d
b d
b
d
a c
a c
a
c
∫ ∫ f ( x, y ) dy dx = ∫ ∫ g ( x ) h ( y ) dy dx = ∫ g ( x ) dx ⋅ ∫ h ( y ) dy
d b
d b
b
d
c a
c a
a
c
∫ ∫ f ( x, y ) dx dy = ∫ ∫ g ( x ) h ( y ) dx dy = ∫ g ( x ) dx ⋅ ∫ h ( y ) dy