(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...

Mathematics
Precalculus
(a) We define the improper integral (over the entire plane

(a) We define the improper integral (over the entire plane R 2 d) where D a is the disk with...

Question:

(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) iswhere S_a is the square with vertices (±a, ±a). Use this to show that

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

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

Transcribed Image Text:

le-42+y?) dA -(x²+y²) -Se-ta?4y?, R2 -(x²+y²) dy dx - 00 -00 -(x²+y²) = lim dA Da e-(x²+y²) dA = T o,

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