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Section 11.1 Problem 11
> N:=64; #other values of N can be used as well. Allow lots of time for
> N=256
N := 64
> delx:=1/N; dely:=1/N; #the x and y - step sizes for the grid
delx := 1/64
dely := 1/64
> f:=(x,y) ->exp(-x^2-y^2);
2 2
f := (x, y) -> exp(-x - y )
> S:=Sum(Sum(f(j*delx+delx/2, k*delx
> +delx/2),k=0..N-1),j=0..N-1)*delx*dely;
S := 1/4096
/ 63 / 63 \\
|----- |----- ||
| \ | \ 2 2 ||
| ) | ) exp(-(1/64 j + 1/128) - (1/64 k + 1/128) )||
| / | / ||
|----- |----- ||
\j = 0 \k = 0 //
> evalf(");
.5577574648
Section 11.3 , Problem 52
> f:=1-x-y; g:=4-x^2-y^2; # Define the two given functions
f := 1 - x - y
2 2
g := 4 - x - y
> p1:=plot3d(f,x=-2..3,y=-2..3): #Assign the plot of f to a variable p1
> p2:=plot3d(g,x=-2..3,y=-2..3): #Assign the plot of g to a variable p2
> with(plots):
> display({p1,p2}); # This command will plot both f and g on the same
> axes.
> eq:=f=g; #Set up the equation to find the curve corresponding to f=g
2 2
eq := 1 - x - y = 4 - x - y
> sol:=solve(eq,y); #Solve this equation for y in terms of x
2 1/2 2 1/2
sol := 1/2 + 1/2 (13 - 4 x + 4 x) , 1/2 - 1/2 (13 - 4 x + 4 x)
# The solution has two branches corresponding to the top
# and bottom half of the circle.
# To set up the integral, we need to find the x-values corresponding to
# where these two brances meet (the left and right points on the
# circle).
> a:=solve(sol[1]=sol[2],x);
1/2 1/2
a := 1/2 + 1/2 14 , 1/2 - 1/2 14
# Now set up the integral. Note that g>f
> integ:=Int(Int(g-f,y=sol[2]..sol[1]),x=a[2]..a[1]);
1/2
1/2 + 1/2 14
/
|
integ := |
|
/
1/2
1/2 - 1/2 14
2 1/2
1/2 + 1/2 (13 - 4 x + 4 x)
/
| 2 2
| 3 - x - y + x + y dy dx
|
/
2 1/2
1/2 - 1/2 (13 - 4 x + 4 x)
> evalf("); #Give this a few minutes to calculate
19.24225500
>