Example 5-6-1 - Maple Help

All Products Maple MapleSim

Home : Support : Online Help : Education : Study Guides : MultivariateCalculus : Chapter 5 : Examples : Section 5-6 : Example 5-6-1

Chapter 5: Double Integration

Section 5.6: Changing Variables in a Double Integral

Example 5.6.1

Let $R$ be the interior and boundary of the triangle whose vertices are $(1, 2)$ , $(4, 9)$ , and $(3, 5)$ .

a)	Integrate $f (x, y) = 2 x + 3 y$ over $R$ , noting that it takes two iterations to cover $R$ .

b)	Make the change of variables $u = (2 y - 3 x - 1) / 5$ , $v = (7 x - 3 y - 1) / 5$ and evaluate the integral of $f$ over the image of $R$ under this change of variables.

Solution

Mathematical Solution

Figure 5.6.1(a) shows the region $R$ ; Figure 5.6.1(b) shows $R'$ , the image of $R$ under the given change of variables.

Figure 5.6.1(a) Region $R$

Figure 5.6.1(b) Region $R'$

Table 5.6.1(a) gives the equations of the edges of the triangle defining region $R$ . Table 5.6.1(b) lists the equations for the mappings between regions $R$ and $R'$ .

Line	Equation
$L_{AB}$	$y = (7 x - 1) / 3$
$L_{BC}$	$y = 4 x - 7$
$L_{CA}$	$y = (3 x + 1) / 2$
Table 5.6.1(a) Edges of $R$

Mapping $R' \to R$	Mapping $R \to R'$
$x = 3 u + 2 v + 1$	$u = (2 y - 3 x - 1) / 5$
$y = 7 u + 3 v + 2$	$v = (7 x - 3 y - 1) / 5$
Table 5.6.1(b) Mappings $R \leftrightarrow R'$

The Jacobian matrix is $[\begin{array}{c} 3 & 2 \\ 7 & 3 \end{array}]$ and the Jacobian itself is $\frac{\partial (x, y)}{\partial (u, v)} = - 5$ , the determinant of the Jacobian matrix.

Since $f (x (u, v), y (u, v)) = 2 (3 u + 2 v + 1) + 3 (7 u + 3 v + 2) = 27 u + 13 v + 8$ , the requisite integrals are then

$\int \int_{R} f (x, y) dy dx$ = $\int_{1}^{3} \int_{\frac{3 x + 1}{2}}^{\frac{7 x - 1}{3}} (2 x + 3 y) &DifferentialD; y &DifferentialD; x + \int_{3}^{4} \int_{4 x - 7}^{\frac{7 x - 1}{3}} (2 x + 3 y) &DifferentialD; y &DifferentialD; x$ = $\frac{160}{3}$

and

$\int \int_{R'} f (x (u, v), y (u, v)) |\frac{\partial (x, y)}{\partial (u, v)}| dv du$ = $\int_{0}^{1} \int_{0}^{1 - u} 5 (27 u + 13 v + 8) &DifferentialD; v &DifferentialD; u$ = $\frac{160}{3}$

Maple Solution - Interactive

Initialize

•	Tools≻Load Package: Student Multivariate Calculus

Loading Student:-MultivariateCalculus

•	Context Panel: Assign Name

$P = [[1, 2], [4, 9], [3, 5]]$ $\overset{assign}{\to}$

•	Context Panel: Assign Name

$f = 2 x + 3 y$ $\overset{assign}{\to}$

•	Context Panel: Assign Name

$U = (2 y - 3 x - 1) / 5$ $\overset{assign}{\to}$

•	Context Panel: Assign name

$V = (7 x - 3 y - 1) / 5$ $\overset{assign}{\to}$

Obtain the equations for the edges of region $R$

•	Make a list of two consecutive vertices. Context Panel: Student Multivariate Calculus≻Lines & Planes≻Line

•	Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻equation

•	Context Panel: Solve≻Isolate Expression for≻ $y$

•	Context Panel: ≻Assign to a Name≻s[12], s[23], s[31] (as appropriate)

$P_{1}, P_{2}$ $\overset{make line}{\to}$ $<< Line 1 >>$ $\overset{representation}{\to}$ $- \frac{7}{3} x + y = - \frac{1}{3}$ $\overset{isolate for y}{\to}$ $y = - \frac{1}{3} + \frac{7}{3} x$ $\overset{assign to a name}{\to}$ $s_{12}$

$P_{2}, P_{3}$ $\overset{make line}{\to}$ $<< Line 2 >>$ $\overset{representation}{\to}$ $- 4 x + y = - 7$ $\overset{isolate for y}{\to}$ $y = - 7 + 4 x$ $\overset{assign to a name}{\to}$ $s_{23}$

$P_{3}, P_{1}$ $\overset{make line}{\to}$ $<< Line 3 >>$ $\overset{representation}{\to}$ $- \frac{3}{2} x + y = \frac{1}{2}$ $\overset{isolate for y}{\to}$ $y = \frac{1}{2} + \frac{3}{2} x$ $\overset{assign to a name}{\to}$ $s_{31}$

Implement the integration over the region $R$

•	Calculus palette: Iterated double-integral template

•	Context Panel: Evaluate and Display Inline

$\int_{1}^{3} \int_{\frac{3 x + 1}{2}}^{\frac{7 x - 1}{3}} f &DifferentialD; y &DifferentialD; x + \int_{3}^{4} \int_{4 x - 7}^{\frac{7 x - 1}{3}} f &DifferentialD; y &DifferentialD; x$ = $\frac{160}{3}$

Change coordinates and implement the integration in the new coordinate system.

Obtain the equations $S = \{x = x (u, v), y = y (u, v)\}$ for the mapping $(u, v) \to (x, y)$

•	Write the equations for the mapping $v (x, t) will now be displayed as v$ . Press the Enter key.

•	Context Panel: Solve≻Solve for Variables≻ $\{x, y\}$

•	Context Panel: Assign to a Name≻ $S$

$u = U, v = V$

$u = \frac{2}{5} y - \frac{3}{5} x - \frac{1}{5}, v = \frac{7}{5} x - \frac{3}{5} y - \frac{1}{5}$

$\overset{solve (specified)}{\to}$

$\{x = 2 v + 1 + 3 u, y = 7 u + 2 + 3 v\}$

$\overset{assign to a name}{\to}$

$S$

Obtain the Jacobian matrix and the Jacobian

•	Expression palette: Evaluation template Evaluate $x$ and then $y$ using the information in set $S$

•	Context Panel: Evaluate and Display Inline

•	Context Panel: Assign to a Name≻ $X$ or $Y$ , as appropriate

•	Form the list $[X, Y]$ Context Panel: Evaluate and Display Inline

•	Context Panel: Student Multivariate Calculus≻Differentiate≻Jacobian

$\binom{x}{} | \binom{}{S}$ = $2 v + 1 + 3 u$ $\overset{assign to a name}{\to}$ $X$

$\binom{y}{} | \binom{}{S}$ = $7 u + 2 + 3 v$ $\overset{assign to a name}{\to}$ $Y$

$[X, Y]$ = $[2 v + 1 + 3 u, 7 u + 2 + 3 v]$ $\overset{Jacobian}{\to}$ $- 5$

Obtain the Jacobian matrix and the Jacobian from first principles

•	Matrix palette: Insert template for $2 \times 2$ matrix.

•	Calculus palette: Partial derivative operator

•	Context Panel: Evaluate and Display Inline

•	Context Panel: Standard Operations≻Determinant

$[\begin{array}{c} \frac{\partial}{\partial u} X & \frac{\partial}{\partial v} X \\ \frac{\partial}{\partial u} Y & \frac{\partial}{\partial v} Y \end{array}]$ = $\overset{determinant}{\to}$ $- 5$

Obtain the images of the edges of the triangle defining region $R$

•	Expression palette: Evaluation template Evaluate the equation of each edge of $R$ using the equations in set $S$ Press the Enter key.

•	Context Panel: Move to Left

•	Context Panel: Solve≻Isolate Expression for≻ $v$ (Just for equation $s_{23}$ .)

$\binom{s_{12}}{} | \binom{}{S}$

$7 u + 2 + 3 v = 2 + \frac{14}{3} v + 7 u$

$\overset{move to left}{\to}$

$- \frac{5}{3} v = 0$

$\binom{s_{23}}{} | \binom{}{S}$

$7 u + 2 + 3 v = - 3 + 8 v + 12 u$

$\overset{move to left}{\to}$

$- 5 u + 5 - 5 v = 0$

$\overset{isolate for v}{\to}$

$v = 1 - u$

$\binom{s_{31}}{} | \binom{}{S}$

$7 u + 2 + 3 v = 2 + 3 v + \frac{9}{2} u$

$\overset{move to left}{\to}$

$\frac{5}{2} u = 0$

Transform the integrand

•	Expression palette: Evaluation template Evaluate $f$ using the equations in set $S$ .

•	Context Panel: Assign to a Name≻ $F$

$\binom{f}{} | \binom{}{S}$ = $13 v + 8 + 27 u$ $\overset{assign to a name}{\to}$ $F$

Implement the integration over the region $R'$

•	Calculus palette: Iterated double-integral template

•	Context Panel: Evaluate and Display Inline

$\int_{0}^{1} \int_{0}^{1 - u} 5 F &DifferentialD; v &DifferentialD; u$ = $\frac{160}{3}$

Maple Solution - Coded

Initialize

•	Install the Student MultivariateCalculus package.

$with (Student :- MultivariateCalculus) &colon;$

•	Define the vertices of the triangle defining $R$ .

$A, B, C ≔ [1, 2], [4, 9], [3, 5] &colon;$

•	Define the integrand $f$ .

$f ≔ (x, y) \to 2 x + 3 y &colon;$

•	Define the mapping $v (x, t) will now be displayed as v$ .

$U ≔ (x, y) \to (2 y - 3 x - 1) / 5 &colon; V ≔ (x, y) \to (7 x - 3 y - 1) / 5 &colon;$

Obtain the equations for the edges of region $R$

•	Use the Line command to form the line object.

•	Apply the GetRepresentation command with the form option set to "equation" to get the equation.

•	Use the isolate command to write the equations in the form $y = \dots$

$s_{AB} ≔ isolate (GetRepresentation (Line (A, B), form = equation), y) &semi; s_{BC} ≔ isolate (GetRepresentation (Line (B, C), form = equation), y) &semi; s_{CA} ≔ isolate (GetRepresentation (Line (C, A), form = equation), y)$

$y = - \frac{1}{3} + \frac{7}{3} x$

Implement the integration over the region $R$

•	Use the Int command to generate the inert integral and apply the value command to evaluate it.

$Q ≔ Int (f (x, y), [y = (3 x + 1) / 2 .. (7 x - 1) / 3, x = 1 .. 3]) + Int (f (x, y), [y = 4 x - 7 .. (7 x - 1) / 3, x = 3 .. 4]) &semi; value (Q)$

$\int_{1}^{3} \int_{\frac{1}{2} + \frac{3}{2} x}^{- \frac{1}{3} + \frac{7}{3} x} (2 x + 3 y) &DifferentialD; y &DifferentialD; x + \int_{3}^{4} \int_{- 7 + 4 x}^{- \frac{1}{3} + \frac{7}{3} x} (2 x + 3 y) &DifferentialD; y &DifferentialD; x$

Change coordinates and implement the integration in the new coordinate system.

Obtain the equations $S = \{x = x (u, v), y = y (u, v)\}$ for the mapping $(u, v) \to (x, y)$

•	Apply the solve command.

$S ≔ solve (\{u = U (x, y), v = V (x, y)\}, \{x, y\})$

$\{x = 3 u + 2 v + 1, y = 7 u + 3 v + 2\}$

Obtain the Jacobian matrix and the Jacobian

•	Use the eval command to obtain a list of expressions $[x (u, v), y (u, v)]$ .

•	Apply the Jacobian command in its two forms, one to get the matrix, and one to get the determinant.

$L ≔ eval ([x, y], S) &semi; Jacobian (L, [u, v]) &semi; Jacobian (L, [u, v], output = determinant)$

$[3 u + 2 v + 1, 7 u + 3 v + 2]$

Obtain the images of the edges of the triangle defining region $R$

•	Use the eval command to make the replacements $x \to x (u, v)$ and $y \to y (u, v)$ in the equation of each edge of the region $R$ .

•	Use the lhs and rhs commands to move all terms to the left of the resulting equation.

$temp ≔ eval (s_{AB}, S) &colon; lhs (temp) - rhs (temp) = 0$

$- \frac{5}{3} v = 0$

$temp ≔ eval (s_{BC}, S) &colon; lhs (temp) - rhs (temp) = 0$

$- 5 u - 5 v + 5 = 0$

Hence: $v = 1 - u$ .

$temp ≔ eval (s_{CA}, S) &colon; lhs (temp) - rhs (temp) = 0$

$\frac{5}{2} u = 0$

Transform the integrand

•	Use the eval command to make the replacements $x \to x (u, v)$ and $y \to y (u, v)$ in $f (x, y)$ .

$F ≔ eval (f (x, y), S)$

$27 u + 13 v + 8$

Implement the integration over the region $R'$

•	Use the Int command to generate the inert integral and apply the value command to evaluate it.

$q ≔ Int (5 F, [v = 0 .. 1 - u, u = 0 .. 1]) &semi; value (q)$

$\int_{0}^{1} \int_{0}^{1 - u} (135 u + 65 v + 40) &DifferentialD; v &DifferentialD; u$

<< Chapter Overview Section 5.6 Next Example >>

© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.

For more information on Maplesoft products and services, visit www.maplesoft.com

Download Help Document

Maple

Maple 附加模块

Student Success Platform

提高存留率

MapleSim

MapleSim 附加模块

系统工程

项目服务

Maple T.A. and Möbius

教育

行业

汽车与航空航天

机器人

机器设计和工业自动化

其他

应用领域

产品价格

购买

院系/全校正版授权

Maplesoft精英维护升级计划

支持

产品培训

产品在线帮助

研讨会与活动

出版

资源中心

示例和应用

社区

关于 Maplesoft

媒体中心

用户社区

联系我们

Online Help

All Products Maple MapleSim

Maple

数学软件

Maple 附加模块

Student Success Platform

提高存留率

MapleSim

多学科系统级建模仿真

MapleSim 附加模块

系统工程

项目服务

Maple T.A. and Möbius

教育

行业

汽车与航空航天

机器人

机器设计和工业自动化

其他

应用领域

产品价格

购买

院系/全校正版授权

Maplesoft精英维护升级计划

支持

产品培训

产品在线帮助

研讨会与活动

出版

资源中心

示例和应用

社区

关于 Maplesoft

媒体中心

用户社区

联系我们

Online Help

All Products Maple MapleSim