(a) We define the improper integral (over the entire plane R²d)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) iswhere 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.)

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,