FORTRAN 90
Lecturer : Rafel Hekmat Hameed
University of Babylon
Subject : Fortran 90
College of Engineering
Year : Second B.Sc.
Mechanical Engineering Dep.

Double Integral
 What is a double integral?
Recall
that
a
single
integral
is
something
of
the
form
௕
න ݂ሺ‫ݔ‬ሻ݀‫ݔ‬
௔
A double integral is something of the form
ඵ ݂ ሺ‫ݔ‬ǡ ‫ݕ‬ሻ݀‫ݕ݀ݔ‬
ோ
Where R is called the region of integration and is a region in the (x,y)
plane. The double integral gives us the volume under the surface z=f(x,y), just
as a single integral gives the area under a curve.
 Evaluation of double integrals
The double integral evaluated as an iterated integral
ௗ ௕
‫ ܫ‬ൌ න න ݂ ሺ‫ݔ‬ǡ ‫ݕ‬ሻ݀‫ݕ݀ݔ‬
௖ ௔
௕ ௗ
ൌ න න ݂ ሺ‫ݔ‬ǡ ‫ݕ‬ሻ݀‫ݔ݀ݕ‬
௔ ௖
We can do the same, using any integration method.
ϭ
Example
Evaluate
ଶǤ଺ ସǤ଺
‫ ܫ‬ൌ න න
ଶ
ͳ
݀‫ݕ݀ݔ‬
‫ݕݔ‬
ସ
Take dx = 0.2 in x-direction,
direction, dy = 0.3 in y-direction. Use Trapezoidal rule in
x- direction and Simpson’s 1/3 rule in y-direction.
Calculate the value of ݂ሺሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ
ଵ
௫௬
Applying trapezoidal rule in x-direction
x
For y=2 : I=0.2/2(0.125+2*0.1191+2*0.1136+0.1087) = 0.0699
For y=2.3 : I=0.2/2(0.1087+2*0.1035+2*0.0988+0.0945)= 0.0608
For y=2.6 : I=0.2/2(0.0962+2*0.0916+2*0.0874+0.0836)=
0.2/2(0.0962+2*0.0916+2*0.0874+0.0836)= 0.0538
We can put in the table
y
2
2.3
2.6
f(y)
0.0699
0.0608
0.0538
Applying Simpson’s 1/3 rule in y-direction
I=0.3/3(0.0699+4*0.0608+0.0538)= 0.0367
Ϯ
program double_integration
implicit none
real,dimension(50)::sum1
integer:: i
,k
real::a , b , n ,dx,dy,f,x,y ,j,sumf,sums
f(x,y)=1/(x*y)
read(*,*) a,b,n,dy
dx=(b-a)/n
j=2
do
k=k+1
sumf=f(a,j)+f(b,j)
do i=1,n-1
x=a+(i*dx)
sumf=sumf+2*f(x,j)
enddo
sumf=sumf*dx/2.0
write(*,40)"for y=",j, "integral=", sumf
40 format(2x,a,f5.2,2x,a,2x,e14.8)
sum1(k)=sumf
if(j.eq.2.6)exit
j=j+dy
enddo
do i=1,k
write(*,10)sum1(i)
10 format(2x,e14.8)
enddo
sums=sum1(1)+sum1(k)
do i=2,k-1
if(i/2*2.eq.i)then
sums=sums+4*sum1(i)
else
ϯ
sums=sums+2*sum1(i)
endif
enddo
sums=sums*dy/3.0
write(*,20)"approximated value of double integration=",sums
20 format(2x,a,e16.8)
end
ϰ