# Mathematics 210 Review - Regions of integration
#
# This worksheet shows how to determine regions
# of integration for double and triple
# integrals.  This worksheet uses a software package
# called CALCPLOT which runs in the
# symbolic manipulation language called Maple.
# You may use it to show that alternate
# descriptions define the same region when the limits
# of integration are changed around.
# The worksheet problems in this sample are from Stewart's
# Calculus Edition II, sections
# 13.4 and 13.7.
#
# NOTE: To invoke the calcplot command displayed in this worksheet,
# place the cursor on the line where the command is written
# and depress the "enter" key on the bottom right corner of
# the keypad (near the number pad.)
#
# To illustrate how to use calcplot, we will completely solve one example.
# We will begin with the command that loads this package in Maple:
> with(calcplot):


# Note:  For an integral stated as  ...dxdy use the calcplot command:
# 'dxdyplot'.
# for an integral stated as ...dydx use the calcplot command: 'dydxplot.'
# Specify the ranges in the same order as the command name, e.g.,
# dydxplot(y=n1..n2,x=m1..m2);  or  dxdyplot(x=m1..m2,y=n1..n2).
#
# We now completely solve the problem 35 on page 804 of Stewart:
#
> Int(Int(f(x,y),y=0..x),x=0..1);
# To draw the integration of integration for this integral, which
# in this case is  0<=x <=1, 0 <= y <=x use the calcplot command
# (which we have included with its output for you to see):
> dydxplot(y=0..x,x=0..1);


# From this picture, we see that the region can also be described as
# 0 <= y <= 1, y <= x <=1
# The calcplot command, with output for drawing
# this region, with the order of  integration interchanged is:
#
> dxdyplot(x=y..1,y=0..1);


# Both drawing cover the same region of the x-y plane, and
# confirm that our two descriptions correspond.  Now let us consider
# problem 36 on page 804:
# 0 <= x <= Pi/2, 0 <= y <= sin(x)
# The calcplot command to draw this region follows.  To invoke it
# yourself use the mouse to p[lace the cursor on the command
# line and hit the ENTER key on the lower right
# corner of the keypad (near the number pad.)
> dydxplot(y=0..sin(x),x=0..Pi/2);
# Interchanging the order of integration with properly chosen limits of
# integration should give the same region.  Please fill in the form given
# below, involve it to graph the region, and check that you have the
# correct answer:
#
> dxdyplot(x=        ,y=          );
# For problem 39 on page 804, 0 <= y <= 4, y/2 <= x <= 2.
# Invoke the calcplot command to draw this region:
> dxdyplot(x=y/2..2,y=0..4);
# What is the corresponding description if we interchange the order of
# integration?  Fill in the values of the calcplot command below and invoke it.
# If the graph covers the same region, you have done it correctly, if not, then
# try again:
#
> dydxplot(y=      ,x=        );
# Sometimes we want to change the order of integration in order to
# make it easier to evaluate the integral.
# Consider the two problems below.  In each case the function
# depends on only one variable,
# while we integrate over two variables.  We can simplify
# the integration by making the variable in the function the same
# as the last integration variable.
# Use calcplot to draw the region of integration and determine how to
# change the order of integration.
#
# From page 804, problem 46:
#
> Int(Int(exp(x^4),x=y^(1/3)..2),x=0..8);
# The region of integration is: 0 <= y <= 8, Y^(1/3) <= x <= 2.
# To draw it, invoke the calcplot command:
#
> dxdyplot(x=y^(1/3)..2,y=0..8);


# From the drawing, try to describe the region in reverse order
# (x as a range of numbers,  y as a specified function of x.)
# Fill in the values for the calcplot command below and invoke it.  If the
# pictures do not match, then try again:
#
> dydxplot(y=        ,x=       );
#
# For problem 42, page 804
> Int(Int(sqrt(1+x^3),x=sqrt(y)..1),y=0..1);
# The region of integration is 0 <= y <= 1, y^(1/2) <= x <= 1.
# To draw it invoke the calcplot command:
#
> dxdyplot(x=y^(1/2)..1,y=0..1);


# From the drawing try to describe the region in reverse order
# (y as a range of  numbers, x as a function of y.)
# Fill in the values for the calcplot command below and invoke it.
# If the pictures do not correspond, try again.
#
> dydxplot(y=    ,x=     );
#
# Now let's consider some regions of integration for 3-variable
# integration.   These problems are taken from Stewart,
# section 13.7, page 829.
# Page 829, problem 23:
#
> Int(Int(Int(1,y=0..2-2*z),z=0..1-x),x=0..1);
# Region of integration: 0 <= x <= 1, 0 <= z <= 1-x, 0 <= y <=2-2*z.
# To draw the region invoke the calcplot command:
#
> dydzdxplot(y=0..2-2*z,z=0..1-x,x=0..1);


# page 829, problem 24:
#
> Int(Int(Int(1,x=0..4-y^2),z=0..2-y),y=0..2);
# Region of integration: 0 <= y <= 2, 0 <= z <= 2-y, ) <= x <= 4-y^2.
# To draw it invoke the calcplot command:
#
> dxdzdyplot(x=0..4-y^2,z=0..2-y, y=0..2);


# Now let's consider changing the order of integration.
# Remember, for this we can use, as the order to
# integrate any of  dzdydx, dzdxdy, dydxdz, dydzdx, dxdydz or dxdzdy.
# Page 829, problem 31:
#
> Int(Int(Int(f(x,y,z),z=0..y),x=y..1),y=0..1);
# Region of integration: 0 <= y <= 1, y <= x <= 1, 0 <= z <= y.
# To draw it, invoke the caclplot command:
#
> dzdxdyplot(z=0..y,x=y..1, y=0..1);


# From the drawing, try to describe the region by using a different order
# of integration. (e.g., x as a range of numbers, y as a function of x, z
# as a function of y and x.)
#
# Fill in the values of the calcplot command below, and invoke it.
# If the regions do not correspond, try again.
#
> dzdydxplot(z=    ,y =        ,x=     );
#
# Page 829, problem 32:
> Int(Int(Int(f(x,y,z),z=0..y),y=0..x^2),x=0..1);
#
# Region of integration: 0 <= x <= 1,  0 <= y <= x^2,  0 <= z <= y.
# To draw it, invoke the calcplot command:
> dzdxdyplot(z=0..y, x=y..1,y=0..1);
#
# From the drawing, try to describe the region by using a different order
# of integration. (e.g., z as a range of numbers, y as a function of z, z
# as a function of y and z.)
#
# Fill in the values of the calcplot command below, and invoke it.
# If the regions do not correspond, try again.
#
> dxdydzplot(x=    ,y =        ,z=     );
#
# Problem 3, page 829:
> Int(Int(Int(xyz,x=0..y),y=0..z),z=0..1);
# Region of integration: 0 <= z <= 1,  0 <= y <= z , 0 <= x <= y.  To draw
# it invoke the calcplot command:
#
> dxdydzplot(x=0..y,y=0..z,z=0..1);
#
# From the drawing, try to describe the region by using a
# different order of integration. (e.g., x as a range of numbers, y
# as a function of x, z as a function of y and x.)
# Fill in the values of the calcplot command below, and invoke it.
# If the regions do not correspond, try again.
#
> dzdydxplot(z=    ,y =        ,x=     );
#
# Problem 4, page 829:
#
> Int(Int(Int(2*x*y,z=0..x+y),y=x..2*x),x=0..1);
#
# Region of integration: 0 <= x <= 1,  0 <= y <= 2*x , 0 <= z <= x+y.
# To draw it invoke the calcplot command:
#
> dzdydxplot(z=0..x+y,y=0..2*x,x=0..1);
#
# From the drawing, try to describe the region by using a different order
# of integration. (e.g., y as a range of numbers, z as a function of y, x
# as a function of y and z.)
#
# Fill in the values of the calcplot command below, and invoke it.
# If the regions do not correspond, try again.
#
> dxdzdxplot(x=     ,z =        ,y =     );