Chapter 6: Applications of Double Integration
Section 6.1: Area
Use the double integral to calculate the area of the region R, the interior of the ellipse x2+4 y2=1.
The region R is shaded in Figure 6.1.8(a). The work of iterating in either order is equivalent; the order selected here is dy dx, resulting in the integral
∫−11∫−1−x2/21−x2/21 dy dx = π2
Note that in Figure 6.1.8(a) the upper branch of the ellipse, given by YT=1−x2/2 is drawn in black, while the lower branch, YB=−YT, is drawn in blue.
Figure 6.1.8(a) The region R
Maple Solution - Interactive
Tools≻Load Package: Student Multivariate Calculus
Access the MultiInt command via the Context Panel
Type the integrand, 1.
Context Panel: Student Multivariate Calculus≻Integrate≻Iterated
Fill in the fields of the two dialogs shown below
Context Panel: Evaluate Integral
The simplest approach is to employ the task template in Table 6.1.8(a).
Calculus - Vector≻Integration≻Multiple Integration≻2-D≻Over an Ellipse
Integrate fx,y over an Ellipse
Equation of ellipse:
From θ= to θ=
Table 6.1.8(a) Task template for integration over an ellipse
Maple elects to implement the integration in polar coordinates, representing the ellipse as
which is what would be obtained by the following "direct" conversion to polar coordinates.
Expression palette: Evaluation template
Context Panel: Solve≻Obtain Solutions for≻r
x2+4 y2=1x=a|f(x)x=r cosθ,y=r sinθ
→solutions for r
A solution from first principles is given in Table 6.1.8(b).
Solve for y=yx
Control-drag (or type) the equation of the ellipse.
Context Panel: Solve≻Obtain Solutions for≻y
Context Panel: Assign to a Name≻Y
x2+4 y2=1→solutions for y−x2+12,−−x2+12→assign to a nameY
Write an appropriate iterated integral and evaluate
Calculus palette: Iterated double-integral template
Context Panel: Evaluate and Display Inline
∫−11∫Y2Y11 ⅆy ⅆx = π2
Table 6.1.8(b) Solution from first principles
Table 6.1.8(c) provides an alternate solution via a visualization task template.
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 6.1.8(c) Solution by visualization task template
Constrained scaling has to be applied to both graphs in Table 6.1.8(c).
Maple Solution - Coded
Install the Student MultivariateCalculus package.
Apply the solve command to the equation of the ellipse.
Top-level, using the Int and int commands
Use the MultiInt command from the Student MultivariateCalculus package
MultiInt1,x,y=Ellipsex2+4 y2=1 = π2
Maple represents the polar form of the ellipse in terms of the tangent function. A simpler representation would be r=1/cos2θ+4 sin2θ, obtained by making the substitutions x=r cosθ,y=r sinθ in x2+4 y2=1.
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