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Department of Mathematics
School of Advanced Sciences
MAT 1011 – Calculus for Engineers (MATLAB)
Experiment 4–A
Double Integrals and change of order of integration
In this experiment, we consider a continuous function f such that f ( x, y)  0 for all

( x, y) in a region R in the xy  plane, then the volume of the solid region that lies above R
and below the graph of f is defined as the double integral V   f ( x, y )dA , where R is the
R

region bounded by the curves y  1 ( x ) and y   2 ( x) between x  a and x  b
...
Hence

V   f ( x, y)dA 
R

b

2 ( x )

 [  f ( x, y)dy]dx

x  a y 1 ( x )

MATLAB Syntax
int(int(f, y, phi1, phi2),x,a,b) where y is the inner variable, x is the outer
variable
...
e
...
Then

V   f ( x, y)dA 
R

d

 2 ( y)

 [  f ( x, y)dx]dy

y c x  1 ( y )

MATLAB Syntax
int(int(f, x, psi1, psi2),y,c,d) where x is the inner variable, y is the outer
variable
...
, viewSolid
...
m
are required
...

Students are advised to upload these files (viewSolid
...
m) to
their MATLAB drive
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Download viewSolid
...
google
...
m from the following link:
https://drive
...
com/file/d/1H1cJOfJArmUujQNVxeSeuGfBJejtbGgJ/view?usp=sharing

Department of Mathematics, Vellore Institute of Technology, Vellore
...

4
1 x/2
2 x

Example
...

4

Example
...
The limits of integration here are y  0 to 1 while x  y to 1
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Department of Mathematics, Vellore Institute of Technology, Vellore
...


Example 3 Evaluate the integral

0 x2

As per the given limits of integration x  0 to 2 while y  x 2 to 2 x
...

y  0 to 4 while x 
2
int(int(4*x+2,x,y/2,sqrt(y)),y,0,4)
ans =
8
Example 4: Evaluate

 ( x  3 y

2

)dA where R  ( x, y ) | 0  x  2, 1  y  2

R

clc
clear all
syms x y z
viewSolid(z, 0+0*x+0*y, x-3*y^2,y,1+0*x, 2+0*x,x,0,2)
int(int(x-3*y^2,y,1,2),x,0,2)
Output:
>> ans
-12
In this figure the required volume is below the
plane z  0 (shown in red) and above the
surface z  x  3y 2 (shown in blue)
...


Department of Mathematics, Vellore Institute of Technology, Vellore
...
The
integral in this example is 0 means V1=V2
Example 6: Find the volume of the solid that lies under the paraboloid z  x 2  y 2 and above
the region D in the xy-plane bounded by the lines y  2 x and the parabola y  x 2
MATLAB Code:
syms x y z
viewSolidone(z, 0+0*x+0*y, x^2+y^2,x,y/2,sqrt(y),y,0,4)
int(int(x^2+y^2,x,y/2,sqrt(y)),y,0,4)
Output:
>>

ans
216/35

Exercise:
1
...

2
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