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Unformatted text preview: Lecture 25 Double Integrals for Non-rectangles In the previous lecture we considered only integrals over rectangular regions. In practice, regions of interest are rarely rectangles and so in this lecture we consider two strategies for evaluating integrals over other regions. Redefining the function One strategy is to redefine the function so that it is zero outside the region of interest, then integrate over a rectangle that includes the region. For example, suppose we need to approximate the value of I = integraldisplayintegraldisplay T sin 3 ( xy ) dx dy where T is the triangle with corners at (0 , 0), (1 , 0) and (0 , 2). Then we could let R be the rectangle [0 , 1] [0 , 2] which contains the triangle T . Notice that the hypotenuse of the triangle has the equation 2 x + y = 2. Then make f ( x ) = sin 3 ( xy ) if 2 x + y 2 and f ( x ) = 0 if 2 x + y > 2. In Matlab we can make this function with the command: > f = inline(sin(x.*y).^3.*(2*x + y <= 2)) In this command <= is a logical command. The term in parentheses is then a logical statement and is given the value 1 if the statement is true and 0 if it is false. We can then integrate the modified f on [0 , 1] [0 , 2] using the command: > I = dblquad(f,0,1,0,2) As another example, suppose we need to integrate x 2 exp( xy ) inside the circle of radius 2 centered...
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