Chapter 6: Applications of Double Integration
Section 6.1: Area
The double integral ∫∫R1 dA will return the area of the region R.
Relevant Maple Tools
The area of the region R bounded below by yBx and above by yTx on the interval x∈a,b can be obtained with the Int and int commands, as shown in Table 6.1.1.
Table 6.1.1 Use of the Int and int commands for iterating a double integral in the order dy dx
The Int command returns the unevaluated integral ∫ab∫yLyT1 dy dx, while the int command immediately evaluates the integral. The form for these two commands as displayed in Table 6.1.1 will be used throughout this manual. It is simpler to use than iterating the commands as in
and it has implications for the numeric evaluation of iterated integrals.
The area of the region R bounded on the left by xLy and on the right by xRy on the interval y∈c,d can be obtained with the Int and int commands, as shown in Table 6.1.2.
Table 6.1.2 Use of the Int and int commands for iterating a double integral in the order dx dy
The Int command returns the unevaluated integral ∫cd∫xLxR1 dx dy, while the int command immediately evaluates the integral.
The Calculus palette contains the template ∫x1x2∫y1y2fⅆyⅆx with which a double integral can be iterated in either order, dy dx or dx dy. This template implements the int command, so even though the integral is displayed, it will evaluate immediately upon accessing it by other operations such as the Context Panel's Simplify. Alternatively, the Context Panel option "Evaluate and Display Inline" will result in evaluation, the effect being the same as that achieved by the Int and int commands as per Tables 6.1.(1, 2).
To obtain the actual unevaluated integral that is equivalent to that set by the Int command, use the Context Panel option 2-D Math≻Convert To≻Inert Form. The integral signs and the "d" operators will change color to gray (from black).
The unevaluated integral can be evaluated via the Context Panel option "Evaluate Integral".
Table 6.1.3 illustrates a task template that not only computes the value of an iterated double integral, but also displays the region over which the integration takes place. This is one of five such task templates designed for integration in Cartesian coordinates (two- and three-dimensional), polar, cylindrical, and spherical coordinates. In the table, it is used to find the area of the right triangle whose legs are along the coordinate axes, and whose hypotenuse is given by the line y=1−x.
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cartesian 2-D
Evaluate ∬RΨx,y dA and Graph R
Area Element dA
Select dAdy dxdx dy
Value of Integral
Table 6.1.3 Visualization task template for two-dimensional Cartesian coordinates
After selecting the order of iteration (here, dy dx), the integrand is set to 1 and the limits of integration are entered. The "Exact" button computes the area of the region displayed in the left-hand graph. The "Draw Graphs" button produces both graphs. The plane region in the left-hand graph uses an arrow to indicate the direction of the inner integral. The right-hand graph shows a solid of height 1 whose volume is the same number as the area of the plane region over which it sits.
Table 6.1.4 illustrates a task template based on the modified int command in the Student VectorCalculus package. This modification recognizes certain pre-defined regions (disk, ellipse, triangle, rectangle) as well as a general region. There is one task template for each of these two-dimensional regions, and a set of templates for three-dimensional regions. The task template in Table 6.1.4 is applied to the right triangle in Table 6.1.3.
Calculus - Vector≻Integration≻Multiple Integration≻2-D≻Over a General 2-D Region
Integrate fx,y over a General Region
∫x=ax=b∫y=uxy=vxf ⅆy ⅆx
Table 6.1.4 Task template for a dy dx iteration over a general region
Unfortunately, this task template supports iteration only in the order dy dx. The underlying int command, detailed in a further subsection, can be made to iterate in either order.
Table 6.1.5 illustrates a task template that implements integration over a triangle defined by its vertices. In the table, the task template is applied to the right triangle in Table 6.1.3.
Calculus - Vector≻Integration≻Multiple Integration≻2-D≻Over a Triangle
Integrate fx,y over a Triangle
Table 6.1.5 Task template for integration over a triangle defined by its vertices
Table 6.1.6 illustrates a task template that implements integration over an ellipse defined by its equation.
Calculus - Vector≻Integration≻Multiple Integration≻2-D≻Over an Ellipse
Integrate fx,y over an Ellipse
Equation of ellipse:
From θ= to θ=
Table 6.1.6 Task template for integration over an ellipse defined by its equation
Maple implements the integration in polar coordinates, and represents the ellipse in these coordinates. Example 6.1.8 shows the relation between the Cartesian and polar representations of an ellipse.
The Student MultivariateCalculus Package
The MultiInt command in the Student MultivariateCalculus package will formulate and evaluate an iterated multiple integral. Its syntax for iteration in the order dy dx is illustrated by the task template in Table 6.1.7. It is applied to the right triangle in Table 6.1.3. Like the task template in Table 6.1.5, it is also restricted to the iteration order dy dx, but the MultiInt command itself can be used to implement any order of iteration (as illustrated in the Examples 6.1.(1 - 8). Access to the MultiInt command is also possible through the Context Panel. See the discussion in, and following, Table 5.3.2.
Calculus - Multivariate≻Integration≻Multiple Integration≻Cartesian 2-D
Iterated Double Integral in Cartesian Coordinates
Inert integral: dy dx
Table 6.1.7 Task template that implements the MultiInt command
If the option "output = integral" is included, the MultiInt command returns the unevaluated integral. If the option "output = steps" is included, the command returns a stepwise evaluation of the integral. If the option "output = value" is included, or if no option at all is included, the integral is evaluated and its value is returned.
Careful inspection of the syntax for the MultiInt command in the task template of Table 6.1.7, and the syntax for the Int and int commands shows that all three use a similar format. All three commands place the bounds for the inner integral first. In other words, for these three commands, write first the integration limits that have to be conceived first, namely, the limits for the inner integral.
As of Maple 2016, the MultiInt command supports integration over the pre-defined regions Circle, Ellipse, Parallelepiped, Rectangle, Region, Section, Sphere, Tetrahedron, and Triangle. These are the same regions supported by the modified int command in the Student VectorCalculus Package.
Syntax for the pre-defined planar regions (Circle, Ellipse, Rectangle, Region, Section, and Triangle) is illustrated in Table 5.3.4.
The Student VectorCalculus Package
Table 6.1.8 lists syntax for various forms of the modified int command in the Student VectorCalculus package. With this command, integration is implemented over several pre-defined regions such as the triangle, ellipse, rectangle, circle, etc.
Table 6.1.8 Instantiations of the modified int command in the Student VectorCalculus package
For the triangle, the vertices are given as vectors.
The ellipse can be defined either by its equation or by its parameters. For the equation, the "= 0" can be omitted. When defining the ellipse by its parameters, the center is given by the vector a,b, the lengths of the semimajor and semiminor axes are given, and the angle by which the ellipse is rotated with respect to the horizontal is given. If the ellipse is not rotated, the angle 0 can be given, or this parameter can be omitted.
If a general region is described in such a way as to conform to iteration in the order dy dx, then the first form given in the table is used. To iterate in the order dx dy, use the second form wherein the variables are reversed as in the list y,x but the positions of the limits remain the same. In other words, the first variable named in the list to the left of the word "Region" must have its range given first. Note well that this order of ranges is the opposite of the order used by the Int, int, and MultiInt commands.
Use the double integral to calculate the area of the region R defined in each of the following examples.
R is the finite region bounded by the graph of y=x 1−x and the x-axis.
R is the finite region bounded by the graphs of x=y2 and y=3−2 x.
R is the region bounded by the graphs of fx=sinx and gx=sin2 x on 0≤x≤π.
R is the region bounded by the graphs of fx=arctanx+1−1/2 and gx=sinx on the interval x∈0,π/2.
R is the interior of the triangle whose vertices are 0,0,a,0,0,b.
R is the interior of the triangle whose vertices are 1,3,7,4,5,9.
R is the region bounded by the graphs of 1, cosx, and y=x on 0≤x≤1.
R is the interior of the ellipse x2+4 y2=1.
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