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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 >>

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