Chapter 5: Double Integration
Section 5.5: Numeric Evaluation of Iterated Integrals
Example 5.5.2
Evaluate ∫0π/2∫x210−x2sinx2+2 x y−y2 ⅆy ⅆx numerically.
Solution
Maple Solution - Interactive
Control-drag the given iterated integral.
Context Panel: 2-D Math≻Convert To≻Inert Form
Context Panel: Approximate≻10 (digits)
∫0π/2∫x210−x2sinx2+2 x y−y2 ⅆy ⅆx→at 10 digits0.2796445908
If the iterated integral is not first converted to its inert form, Maple will immediately seek an exact evaluation of the integral. When it discovers that it cannot find the appropriate antiderivative(s), it then switches to numeric mode. Since it can take Maple quite some time before it gives up on the search for an exact value, if the user knows in advance that a numeric evaluation is called for, it should be initiated immediately by first converting the integral to inert form.
For some iterated double integrals, an exact evaluation of the inner integral is possible, as it is here.
Alternate evaluation
Control-drag the given integral and press the Enter key.
∫0π/2∫x210−x2sinx2+2 x y−y2 ⅆy ⅆx
∫012π2πFresnelSxx−12πcosx22−122πFresnelSxx−12π−2πFresnelCxx−12πsinx2cosx2+2πFresnelSx2+x−102πcosx22−122πFresnelSx2+x−102π−2πFresnelCx2+x−102πsinx2cosx2ⅆx
→at 10 digits
0.2796445909
The Fresnel functions resulting from the inner integration are special functions known to Maple. The outer integral must then be implemented numerically.
Note: The French name "Fresnel" can be pronounced "Frěn-něl" with the accent on the second syllable, or "Frāy-něl" with the accent on the first. The "s" is not pronounced. (The internet has a number of sites where the proper pronunciation can be heard.)
Maple Solution - Coded
Use the Int command to form the unevaluated (inert) integral.
q≔Intsinx2+2 x y−y2,y=x2..10−x2,x=0..π/2:
Apply the evalf command to launch a numeric evaluation of the integral.
evalfq = 0.2796445908
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