Chapter 5: Double Integration
Section 5.2: Iterated Double Integrals
Example 5.2.3
If fx,y=1+5 x2+7 y2, and R is the rectangular region −1≤x≤1,−2≤y≤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=−2y=21+5 x2+7 y2 ⅆy ⅆx = 96
∫y=−2y=2∫x=−1x=11+5 x2+7 y2 ⅆx ⅆy = 96
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∫−221+5 x2+7 y2 dy dx, hold x constant and integrate 1+5 x2+7 y2 with respect to y to get y+5 x2y+7 y3/3y=−2y=2=20 x2+124/3. Integrated this with respect to x, obtaining 20 x3/3+124 x/3x=−1x=1=96.
To evaluate the iterated integral ∫−22∫−111+5 x2+7 y2 dx dy, hold y constant and integrate 1+5 x2+7 y2 with respect to x to get x+5 x3/3+7 x y2x=−1x=1=14 y2+16/3. Integrated this with respect to y, obtaining 14 y3/3+16 y/3y=−2y=2=96.
Maple Solution - Interactive
Initialize
Context Panel: Assign Name
f=1+5 x2+7 y2→assign
Iterate and evaluate
Calculus palette: Template for (definite) iterated double-integral
Context Panel: Evaluate and Display Inline
∫−11∫−22f ⅆy ⅆx = 96
∫−22∫−11f ⅆx ⅆy = 96
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∫−22f ⅆy ⅆx
∫−11∫−225x2+7y2+1ⅆyⅆx
=
96
∫−22∫−11f ⅆx ⅆy
∫−22∫−115x2+7y2+1ⅆxⅆy
Maple Solution - Coded
Define the integrand.
f≔1+5 x2+7 y2:
Use the Int command to set the unevaluated integral, and the int command for immediate evaluation.
Intf,x=−1..1,y=−2..2=intf,x=−1..1,y=−2..2
∫−22∫−115x2+7y2+1ⅆxⅆy=96
Intf,y=−2..2,x=−1..1=intf,y=−2..2,x=−1..1
∫−11∫−225x2+7y2+1ⅆyⅆx=96
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=−2..2,output=integral
MultiIntf,y=−2..2,x=−1..1,output=integral
Use the MultiInt command to return the value of the iterated integral
MultiIntf,x=−1..1,y=−2..2 = 96
MultiIntf,y=−2..2,x=−1..1 = 96
Use the MultiInt command to obtain a stepwise evaluation of the iterated integral
MultiIntf,x=−1..1,y=−2..2,output=steps
∫−22∫−115x2+7y2+1ⅆxⅆy=∫−2253x3+7y2x+xx=−1..1|53x3+7y2x+xx=−1..1ⅆy=∫−22163+14y2ⅆy=163y+143y3y=−2..2|163y+143y3y=−2..2
MultiIntf,y=−2..2,x=−1..1,output=steps
∫−11∫−225x2+7y2+1ⅆyⅆx=∫−1173y3+5x2y+yy=−2..2|73y3+5x2y+yy=−2..2ⅆx=∫−1120x2+1243ⅆx=203x3+1243xx=−1..1|203x3+1243xx=−1..1
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