MATH 251: Calculus 3, SET8

15: Multiple Integrals

15.1: Double Integrals Over Rectangles

These problems are done with the CAS. See Hand Solutions for details.

1. [999/10]

With a rectangular region of integration, we can always easily change the order of integration since the limits of integration are constants. It may be the case that an integral is harder (or even impossible) to integrate analytically (by hand or with a CAS) in a particular order. So be open to changing the order of integation if needed.
syms x y
Ixy = int(int(2*x+1, x, 0, 2), y, 0, 4) % The result
Ixy = 24
Iyx = int(int(2*x+1, y, 0, 4), x, 0, 2) % is the same.
Iyx = 24

2. [999/14]

syms x y
f(x,y) = y*sqrt(x+2)
f(x, y) = 
I1 = expand( int(f(x,y), x, 0, 2) )
I1 = 
I2 = int(f(x,y), y, 0, 3)
I2 = 

3. [1000/18]

Here we additionally approximate the result after computing the exact value.
syms x y
I = expand( int(int(sin(x) + sin(y), y, 0, pi/2), x, 0, pi/6) )
I = 
I_appx = double(I)
I_appx = 0.7340

4. [1000/22]

An approximate answer is often useful in scientific and engineering work.
I = int(int(y*exp(x-y), x, 0, 2), y, 0, 1)
I = 
I_appx = double(I)
I_appx = 1.6883

5. [1000/26]

syms s t
I = int(int(sqrt(s+t), s, 0, 1), t, 0, 1)
I = 
I_appx = double(I)
I_appx = 0.9752

6. [1000/30]

syms t theta
I = int(int(tan(theta) / sqrt(1-t^2), t, 0, 1/2), theta, 0, pi/3)
I = 
I_appx = double(I)
I_appx = 0.3629

7. [1000/34]

syms x y
I = int(int(1/(1+x+y), x, 1, 3), y, 1, 2)
I = 
I_appx = double(I)
I_appx = 0.4540

8. [1000/40]

syms x y
z1 = x^2 + x*y^2, z2 = 0 % Curved surface is above xy-plane (z=0).
z1 = 
z2 = 0
V = int(int(z1-z2, x, 0, 5), y, -2, 2)
V = 
V_appx = double(V) % in cm^3
V_appx = 233.3333

9. [1000/44]

syms x y
z1 = 2*x*y / (x^2+1), z2 = x + 2*y
z1 = 
z2 = 
figure
fsurf(z1, [0 2 0 4], 'g', 'MeshDensity', 12); hold on
fsurf(z2, [0 2 0 4], 'r', 'MeshDensity', 12)
view(130,20); xlabel('x'); ylabel('y'); zlabel('z')
xticks(0:2); yticks(0:2:4); zticks(0:2:10)
title('SET8, 1000/44: Plane is above curved surface.')
%
V = int(int(z2-z1, x, 0, 2), y, 0, 4)
V = 
V_appx = double(V) % in cm^3
V_appx = 27.1245

10. [1000/48]

The average value of f over a region R is given by .
syms x y
f(x,y) = exp(y)*sqrt(x+exp(y))
f(x, y) = 
I = int(int(f(x,y), y, 0, 1), x, 0, 4)
I = 
A = 4*1
A = 4
f_ave = I/A
f_ave = 
f_ave_appx = double(f_ave)
f_ave_appx = 3.3270
%
figure
fsurf(f(x,y), [0 4 0 1], 'c', 'MeshDensity', 12); hold on
fsurf(f_ave, [0 4 0 1], 'y', 'MeshDensity', 12)
xlabel('x'); ylabel('y'); zlabel('z'); alpha 0.7
xticks(0:4); yticks(0:0.5:1); zticks(1:2:7)
title('SET8, 1000/48') % The average value represents the average height of the surface.