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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 37 "Double Integrals in Polar
 Coordinates" }}{PARA 0 "" 0 "" {TEXT -1 14 "Type I. dr dt " }}{PARA 
0 "" 0 "" {TEXT -1 97 "Find the area of the region bounded by the y-ax
is, y=x and the unit circle in the first quadrant." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "plot(\{sqrt(1-x^2),-sqrt(1-x^2),x\},x=-1..2,y=-1..2,t
hickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "with(studen
t): A:=Doubleint(r,r=0..1,t=Pi/4..Pi/2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "value(A);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "with(s
tudent): B:=Doubleint(r,t=Pi/4..Pi/2,r=0..1);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "value(B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 
"Remark: In this case, there is no difference between drdt and dtdr." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "Exampl
e 2. Find the area bounded by the unit circle of radius 1 with center \+
(0,1) and the line  y =x. [Verify that this area can be found by using
 drdt.]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot(\{sqrt(1-x^
2)+1,-sqrt(1-x^2)+1,x\},x=0..2,thickness=2);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 43 "C:=Doubleint(r,r=0..2*sin(t),t=Pi/4..Pi/2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(C);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 15 "Type II. dt dr " }}{PARA 0 "" 0 "" {TEXT -1 108 "
Example 3. Find the area bounded by the curve r =t^2, the unit circel \+
and the y-axis in the first quadrant. " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 100 "plot(\{[cos(t),sin(t),t=0..2*Pi],[t^2*cos(t),t^2*sin
(t),t=0..Pi/2]\},scaling=constrained,thickness=2);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 37 "polarplot(\{3*cos(t),cos(t)\},t=0..Pi);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "E:=Doubleint(r,t=sqrt(r)..P
i/2,r=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(E);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "Example 4. page 667, #19. Find \+
the volume above the cone z =sqrt(x^2+y^2) and below the sphere x^2 +y
^2 +z^2 =1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "implicitplo
t3d(\{z=sqrt(x^2+y^2),x^2 +y^2 +z^2 =1\},x=-3..3,y=-3..3,z=-3..3,numpo
ints=10000,axes=boxed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}
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