(a) We define the improper integral (over the entire plane R 2 d) where D a is the disk with radius a and center the origin Show that (b) An equivalent definition of the improper integral in part (a) is where S a is the square with vertices (a, a) Use this to show that (c) Deduce that (d) By making the change of variable t 2 x, show that (This is a fundamental result for probability and statistics ) le 42 y ) dA (x y) Se ta 4y , R2 (x y) dy dx 00 00 (x y) lim dA Da e (x y) dA T o,

Question: (a) We define the improper integral (over the entire plane R 2 d) where D a is the disk with radius a and center the

(a) We define the improper integral (over the entire plane R²d) le-42+y?) dA -(x²+y²) -Se-ta?4y?, R2 -(x²+y²) dy dx - 00 -00 -(x²+y²) = lim dA Da where D_a is the disk with radius a and center the origin. Show that

e-(x²+y²) dA = T o,

(b) An equivalent definition of the improper integral in part (a) is le-42+y?) dA -(x+y) -Se-ta?4y?, R2 -(x+y) dy dx - 00 -00 -(x+y) where S_a is the square with vertices (±a, ±a). Use this to show that

= lim dA Da e-(x+y) dA = T o, (c) Deduce that

(d) By making the change of variable t = √2 x, show that

(This is a fundamental result for probability and statistics.)

le-42+y?) dA -(x+y) -Se-ta?4y?, R2 -(x+y) dy dx - 00 -00 -(x+y) = lim dA Da e-(x+y) dA = T o,

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