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Chapter 5: Double Integration

Section 5.3: Regions with Curved Boundaries

Essentials

A double integral of $f (x, y)$ over a plane region that can be described by the inequalities $a \leq x \leq b, y_{B} (x) \leq y \leq y_{T} (x)$ can be iterated as

$\int_{a}^{b} \int_{y = y_{B} (x)}^{y = y_{T} (x)} f (x, y) dy dx$

Figures 5.3.(1, 2) show some of the regions for which this holds.

use plots,Student:-VectorCalculus in
module()
local p1,p2,p3,p4,p5,V,q,YT,YB;
YT:=3+(2/3)*sqrt(-x^2+8*x-7):
YB:=3-(2/3)*sqrt(-x^2+8*x-7):
V:=RootedVector(root=[4,1],<0,4>):
p1:=shadebetween(YB,YT,x=1..7,scaling=constrained,view=[0..7,0..5],color=red):
p2:=plot([YT,YB],x=1..7,color=[black,green],thickness=3):
p3:=textplot({[1.7,4.9,typeset(y[T])],[1.7,1.2,typeset(y[B])],[1.5,.3,typeset(x=a)],[6.5,.3,typeset(x=b)]}):
p4:=plot([[[1,0],[1,3]],[[7,3],[7,0]]],linestyle=dot,color=blue):
p5:=PlotVector(V,color=black):
q:=display(p5,p1,p2,p3,p4,labels=[x,y],tickmarks=[0,0]);
print(q);
end module:
end use:

Figure 5.3.1

use plots, Student:-VectorCalculus in
module()
local p1,p2,p3,p4,p5,p6,q,V,YB,YT;
V:=RootedVector(root=[4,1],<0,7>):
YT:=6+(2/3)*sqrt(-x^2+8*x-7):
YB:=3-(2/3)*sqrt(-x^2+8*x-7):
p1:=shadebetween(YB,YT,x=1..7,scaling=constrained,view=[0..7,0..8],color=red):
p2:=plot([YT,YB],x=1..7,color=[black,green],thickness=3):
p3:=textplot({[1.7,7.9,typeset(y[T])],[1.7,1.2,typeset(y[B])],[1.5,.3,typeset(x=a)],[6.5,.3,typeset(x=b)]}):
p4:=plot([[[1,3],[1,6]],[[7,3],[7,6]]],linestyle=dot,thickness=5,color=gold):
p5:=plot([[[1,0],[1,3]],[[7,3],[7,0]]],linestyle=dot,color=blue):
p6:=PlotVector(V,color=black):
q:=display(p6,p1,p2,p3,p4,p5,labels=[x,y],tickmarks=[0,0]);
print(q);
end module:
end use:

Figure 5.3.2

A double integral of $f (x, y)$ over a plane region that can be described by the inequalities $x_{L} (y) \leq x \leq x_{R} (y), c \leq y \leq d$ can be iterated as

$\int_{c}^{d} \int_{x = x_{L} (y)}^{x = x_{R} (y)} f (x, y) dx dy$

Figures 5.3.(3,4) show some of the regions for which this holds.

use plots, Student:-VectorCalculus in
module()
local XR,XL,V,q,p1,p2,p3,p4,p5,p6;
V:=RootedVector(root=[1,4],<4,0>):
XR,XL:=3+(2/3)*sqrt(-y^2+8*y-7), 3-(2/3)*sqrt(-y^2+8*y-7):
p1:=implicitplot(9*(x-3)^2+4*(y-4)^2 <= 36,x=0..6,y=0..8,filledregions=true,scaling=constrained,view=[0..6,0..8]):
p2:=implicitplot(x=XL,x=1..3,y=1..7,color=green,thickness=3,gridrefine=3):
p3:=implicitplot(x=XR,x=3..5,y=1..7,color=black,thickness=3,gridrefine=3):
p4:=plot([[[0,7],[3,7]],[[3,1],[0,1]]],linestyle=dot,color=black):
p5:=PlotVector(V,color=black):
p6:=textplot({[1,7.5,typeset(y=d)],[1,.5,typeset(y=c)],[1,6.,typeset(x[L])],[5,2,typeset(x[R])]}):
q:=display(p5,p1,p2,p3,p4,p6,labels=[x,y],tickmarks=[0,0]):
print(q);
end module:
end use:

Figure 5.3.3

use plots, Student:-VectorCalculus, plottools in
module()
local p1,p2,p3,p4,p5,p6,q,V,YB,YT,p124,pp;
V:=RootedVector(root=[1,4],<7,0>):
YT:=6+(2/3)*sqrt(-x^2+8*x-7):
YB:=3-(2/3)*sqrt(-x^2+8*x-7):
p1:=shadebetween(YB,YT,x=1..7,scaling=constrained,color=red):
p2:=plot([YT,YB],x=1..7,color=[black,green],thickness=3):
p4:=plot([[[1,3],[1,6]],[[7,3],[7,6]]],linestyle=dot,thickness=5,color=gold):
p124:=display(p1,p2,p4):
pp:=rotate(p124,-Pi/2,[4,4]):
p3:=textplot({[1.,6,typeset(x[L])],[7.7,1.2,typeset(x[R])],[1,.5,typeset(y=c)],[1,7.5,typeset(y=d)]}):
p5:=plot([[[0,1],[3,1]],[[3,7],[0,7]]],linestyle=dot,color=blue):
p6:=PlotVector(V,color=black):
q:=display(p6,pp,p3,p5,labels=[x,y],tickmarks=[0,0],view=[0..9,0..8]);
print(q);
end module:
end use:

Figure 5.3.4

Iterated Integrals in Maple

Table 5.3.1 lists two ways to iterate a double integral of $f (x, y)$ at top level. It uses either the int or Int commands, the first leading to immediate evaluation; the second, to the unevaluated (or inert) integral.

$\int_{x = a}^{x = b} \int_{y = y_{B} (x)}^{y = y_{T} (x)} f (x, y) dy dx$

$Int (f (x, y), [y = y_{B} (x) .. y_{T} (x), x = a .. b]) = int (f (x, y), [y = y_{B} (x) .. y_{T} (x), x = a .. b])$

$Int (Int (f (x, y), y = y_{B} (x) .. y_{T} (x)), x = a .. b) = int (int (f (x, y), y = y_{B} (x) .. y_{T} (x)), x = a .. b)$

$\int_{y = c}^{y = d} \int_{x = x_{L} (y)}^{x = x_{R} (y)} f (x, y) dx dy$

$Int (f (x, y), [x = x_{L} (y) .. x_{R} (y), y = c .. d]) = int (f (x, y), [x = x_{L} (y) .. x_{R} (y), y = c .. d])$

$Int (Int (f (x, y), x = x_{L} (y) .. x_{R} (y)), y = c .. d) = int (int (f (x, y), x = x_{L} (y) .. x_{R} (y)), y = c .. d)$

Table 5.3.1 Iterating a double integral with top-level commands

The first syntax wherein a single call to either Int or int is made with a list as second argument is the preference throughout this Study Guide. Unfortunately, the iterated double-integral template in the Calculus palette is still a place-holder for the iteration of the int command.

Table 5.3.2 details the MultiInt command in the Student MultivariateCalculus package.

•

The MultiInt command in the Student MultivariateCalculus package behaves much like the preferred syntax for the Int command, namely, that the arguments after the integrand need only be equations connecting a variable name with a range. These equations are not placed in a list, but the order in which they are to be given is the same as for the Int command.

•

After the equations specifying the region of integration, the MultiInt command can take the additional parameters output = integral or output = steps. The first such parameter causes the command to return the unevaluated integral; the second, the detailed steps by which the integral is evaluated. The option output = value is the default, and it is not needed to obtain the immediate evaluation of the integral.

•

The MultiInt command itself has been updated to recognize the same pre-defined domains that the modified int command in the VectorCalculus packages have recognized since their inception. The two-dimensional domains are Circle, Ellipse, Rectangle, Sector, and Triangle; the three-dimensional, are Parallelepiped, Sphere, and Tetrahedron. The Region option is used to define an arbitrary region in either two or three dimensions. The syntax for implementing integration over one of these domains is that of the modified int command in the VectorCalculus packages: [vars] = Domain. See Table 5.3.4 for details.

•	There is access to the MultiInt command via the Context Panel once the Student MultivariateCalculus package has been loaded. For example, bring up the Context Panel on $f (x, y)$ , an expression in $x$ and $y$ , to serve as the integrand, and follow the path

Student Multivariate Calculus≻Integrate≻Iterated

The pane shown in Figure 5.3.1 appears and is used to fix the coordinate system and the names of the variables in that system. Clicking the OK button launches the pane shown in Figure 5.3.2 where the order of iteration is established, and the limits of integration imposed. The drop-down box to the right of "Output" provides three options, namely, the value of the integral, the inert integral (shown to the right of Figure 5.3.2), and a stepwise evaluation of the iterated integral.

Unfortunately, Context Panel access to the MultiInt command does not include access to the updated recognition of pre-defined domains.

Table 5.3.2 The MultiInt command in the Student MultivariateCalculus package

Figure 5.3.1 Coordinates

Figure 5.3.2 Limits and output

$\int_{a}^{b} \int_{u (x)}^{v (x)} f (x, y) dy dx$

Table 5.3.3 contains the visualization task template that not only evaluates an iterated double integral, but also displays the planar region over which the integral takes place. In Table 5.3.3, the double integral of $f (x, y) = x y$ is integrated over the region bounded by $y = x^{2}$ and $y = \sqrt{x}$ . The left-hand graph shows the region, and the arrow guides the eye in visualizing that the first (inner) integration is in the $y$ -direction. The right-hand graph shows the integrand over the region of integration. If $f (x, y) \geq 0$ , then this graph shows the solid whose volume is computed by the iterated integral.

The display in the table shows the end-state of the task template. To re-initialize it so that, for example, the buttons under "Value of Integral" work, re-select the Area Element. (If one of these buttons is pressed first, the Area Element will reset to "Select dA.")

Tools≻Tasks≻Browse:

Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cartesian 2-D

Evaluate $\iint_{R} Ψ (x, y) dA$ and Graph $R$

Area Element $dA$

, $Ψ =$

Value of Integral

$G =$	$b =$
$g =$	$a =$

Bounding Curves

"Volume"

Table 5.3.3 Cartesian 2-D visualization task-template

Table 5.3.4 lists examples of the correct syntax for the two-dimensional domains recognized by the MultiInt command. In each case, the optional parameter output = integral causes the command to return the unevaluated integral.

Region	Syntax	Comments
Circle	$MultiInt (f (x, y), [x, y] = Circle (⟨a, b⟩, r))$	Center: $⟨a, b⟩$ ; radius: $r$
Ellipse - equation	$MultiInt (f (x, y), [x, y] = Ellipse (a x^{2} + b y^{2} = c))$	$a, b, c$ positive
Ellipse - parameters	$MultiInt (f (x, y), [x, y] = Ellipse (⟨u, v⟩, a, b, θ))$	Center: $⟨u, v⟩$ ; lengths of semimajor and semiminor axes: $a, b$ ; Angle of rotation: $θ$ (optional)
Sector - Circle	$MultiInt (f (x, y), [x, y] = Sector (Circle (⟨a, b⟩, r), α, β))$	Start angle: $α$ ; End angle: $β$
Sector - Ellipse	$MultiInt (f (x, y), [x, y] = Sector (Ellipse (⟨u, v⟩, a, b), α, β))$	Start angle: $α$ ; End angle: $β$
Rectangle	$MultiInt (f (x, y), [x, y] = Rectangle (a .. b, c .. d))$	$a \leq x \leq b, c \leq y \leq d$
Triangle	$MultiInt (f (x, y), [x, y] = Triangle (v_{1}, v_{2}, v_{3}))$	$v_{1}, v_{2}, v_{3}$ are the vertices of the triangle, given as vectors.
Region ( $dy dx$ )	$MultiInt (f (x, y), [x, y] = Region (a .. b, y_{B} (x) .. y_{T} (x)))$	$\int_{a}^{b} \int_{y_{B} (x)}^{y_{T} (x)} f (x, y) dy dx$
Region ( $dx dy$ )	$MultiInt (f (x, y), [y, x] = Region (c .. d, x_{L} (y) .. x_{R} (y)))$	$\int_{c}^{d} \int_{x_{L} (y)}^{x_{R} (y)} f (x, y) dx dy$
Table 5.3.4 Regions understood by the MultiInt command in the Student MultivariateCalculus package

Note the behavior of the Region option for integrating over a general region. The order of the integration is controlled by the list of variable names that follows the integrand. The order of the ranges is precisely the order of the ranges as they appear on the iterated integrals. This is the opposite of the behavior for Int and MultiInt where the range for the inner integral is always given first!

Tables 5.3.(5 - 9) contain examples of task-templates that are based on the commands in Table 5.3.4.

Tools≻Tasks≻Browse:

Calculus - Vector≻Integration≻Multiple integration≻2D≻

The center of the disk is given in Cartesian coordinates.

Integrate $f$ over a Disk
$\begin{array}{c} f (x, y) \\ or \\ f (r, θ) \end{array}} =$
Center $(h, k)$ : $h =$ , $k =$ Radius =
From $θ =$ to $θ =$
$d θ dr$

Table 5.3.5 Task template for integration over a disk

Tools≻Tasks≻Browse:

Calculus - Vector≻Integration≻Multiple integration≻2D≻

Integrate $f (x, y)$ over an Ellipse
$f (x, y) =$
Equation of ellipse:
From $θ =$ to $θ =$

Table 5.3.6 Task template for integration over an ellipse

Tools≻Tasks≻Browse:

Calculus - Vector≻Integration≻Multiple integration≻2D≻

Integrate $f (x, y)$ over a Rectangle
$f (x, y) =$
Inner Integral: $\leq x \leq$ Outer Integral: $\leq y \leq$
$dx dy$

Table 5.3.7 Task template for integration over a rectangle

Tools≻Tasks≻Browse:

Calculus - Vector≻Integration≻Multiple integration≻2D≻

Integrate $f (x, y)$ over a Triangle

$f (x, y) =$

Vertices	$x_{1} =$	$x_{2} =$	$x_{3} =$
	$y_{1} =$	$y_{2} =$	$y_{3} =$

$dy dx$

Table 5.3.8 Task template for integration over a triangle

Tools≻Tasks≻Browse:

Calculus - Vector≻Integration≻Multiple integration≻2D≻

Integrate $f (x, y)$ over a General Region

$f (x, y) =$

$\int_{x = a}^{x = b} \int_{y = u (x)}^{y = v (x)} f &DifferentialD; y &DifferentialD; x$

$v (x) =$

$b =$

$u (x) =$

$a =$

$dy dx$

Table 5.3.9 Task template for integration over a general 2-D region

Examples

In Examples 5.3.(1 - 10), integrate $f (x, y) = 1 + 2 x^{2} + 3 y^{2}$ over the given region $R$ .

Example 5.3.1	$R = {(x, y) \| 0 \leq x \leq 1, x^{2} \leq y \leq \sqrt{x}}$
Example 5.3.2	$R = {(x, y) \| - 2 \leq x \leq 3, 0 \leq y \leq 6 + x - x^{2}}$
Example 5.3.3	$R$ is the interior and boundary of the triangle whose vertices are $(0, 0), (3, 1), (2, 5)$ .
Example 5.3.4	$R = {(x, y) \| 0 \leq x \leq 1, x^{3} / 2 \leq y \leq 2 - \sqrt{2 - x^{2}}}$
Example 5.3.5	$R = {(x, y) \| \sqrt{1 - y^{2}} \leq x \leq 4 - \sqrt{1 - y^{2}}, - 1 \leq y \leq 1}$
Example 5.3.6	$R = {(x, y) \| - 3 \leq x \leq 2, x^{2} - 6 \leq y \leq - x}$
Example 5.3.7	$R$ is the interior and boundary of the ellipse $4 x^{2} + 9 y^{2} = 36$ .
Example 5.3.8	$R$ is the region bounded by the ellipse $4 x^{2} + 9 y^{2} = 36$ and the line $y = 2 - 2 x / 3$ .
Example 5.3.9	$R$ is the region bounded by the ellipse $4 x^{2} + 9 y^{2} = 36$ and the line $y = 2 - (1 + \sqrt{5} / 3) x$ .
Example 5.3.10	$R = {(x, y) \| 0 \leq x \leq π, \cos (x) \leq y \leq 1 + \sin (x)}$

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