Chapter 5: Double Integration
Section 5.2: Iterated Double Integrals
Example 5.2.2
If fx,y=7−3 x2−5 y2, and R is the rectangular region −1≤x≤1,−1/2≤y≤1/2, evaluate ∫∫Rf dA by both possible iterations.
Solution
Mathematical Solution
The two possible iterations of the double integral evaluated in Example 5.1.1 are
∫x=−1x=1∫y=−1/2y=1/27−3 x2−5 y2 ⅆy ⅆx = 676
∫y=−1/2y=1/2∫x=−1x=17−3 x2−5 y2 ⅆx ⅆy = 676
The "inner" integral is evaluated first, holding the "other" variable constant. The result is then a single definite integral, that is, the "outer" integral is the integral of the result of the "inner" integral.
To evaluate the iterated integral ∫−11∫−1/21/27−3 x2−5 y2 dy dx, hold x constant and integrate 7−3 x2−5 y2 with respect to y to get 7 y−3 x2y−5 y3/3y=−1/2y=1/2=79/12−3 x2. Integrated this with respect to x, obtaining 79 x/12−x3x=−1x=1=67/6.
To evaluate the iterated integral ∫−1/21/2∫−117−3 x2−5 y2 dx dy, hold y constant and integrate 7−3 x2−5 y2 with respect to x to get 7 x−x3−5 x y2x=−1x=1=12−10 y2. Integrated this with respect to y, obtaining 12 y−10 y3/3y=−1/2y=1/2=67/6.
Maple Solution - Interactive
Initialize
Context Panel: Assign Name
f=7−3 x2−5 y2→assign
Iterate and evaluate
Calculus palette: Template for (definite) iterated double-integral
Context Panel: Evaluate and Display Inline
∫−11∫−1/21/2f ⅆy ⅆx = 676
∫−1/21/2∫−11f ⅆx ⅆy = 676
Inert iterated double-integral and its evaluation
Calculus palette: Template for (definite) iterated double-integral Context Panel: 2-D Math≻Convert To≻Inert Form Press the Enter key.
Context Panel: Evaluate Integral
∫−11∫−1/21/2f ⅆy ⅆx
∫−11∫−1212−3x2−5y2+7ⅆyⅆx
=
676
∫−1/21/2∫−11f ⅆx ⅆy
∫−1212∫−11−3x2−5y2+7ⅆxⅆy
Maple Solution - Coded
Define the integrand.
f≔7−3 x2−5 y2:
Use the Int command to set the unevaluated integral, and the int command for immediate evaluation.
Intf,x=−1..1,y=−1/2..1/2=intf,x=−1..1,y=−1/2..1/2
∫−1212∫−11−3x2−5y2+7ⅆxⅆy=676
Intf,y=−1/2..1/2,x=−1..1=intf,y=−1/2..1/2,x=−1..1
∫−11∫−1212−3x2−5y2+7ⅆyⅆx=676
The MultiInt command in the Student MultivariateCalculus package is designed to set up an iterated multiple integral, and return its value, its inert form, or a stepwise evaluation. Section 5.3 details how this command can be accessed through the Context Panel.
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Use the MultiInt command to return the unevaluated iterated integral
MultiIntf,x=−1..1,y=−1/2..1/2,output=integral
MultiIntf,y=−1/2..1/2,x=−1..1,output=integral
Use the MultiInt command to return the value of the iterated integral
MultiIntf,x=−1..1,y=−1/2..1/2 = 676
MultiIntf,y=−1/2..1/2,x=−1..1 = 676
Use the MultiInt command to obtain a stepwise evaluation of the iterated integral
MultiIntf,x=−1..1,y=−1/2..1/2,output=steps
∫−1212∫−11−3x2−5y2+7ⅆxⅆy=∫−1212−x3−5xy2+7xx=−1..1|−x3−5xy2+7xx=−1..1ⅆy=∫−1212−10y2+12ⅆy=−103y3+12yy=−12..12|−103y3+12yy=−12..12
MultiIntf,y=−1/2..1/2,x=−1..1,output=steps
∫−11∫−1212−3x2−5y2+7ⅆyⅆx=∫−11−53y3−3x2y+7yy=−12..12|−53y3−3x2y+7yy=−12..12ⅆx=∫−11−3x2+7912ⅆx=−x3+7912xx=−1..1|−x3+7912xx=−1..1
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