Quiz/worksheet #7 MAT 309-1
Name........................................
Please show ALL of your work clearly and neatly on separate paper(s). They will NOT be
accepted unless they are stapled to this cover sheet. No work = No credit.
1. (a) We define the improper integral (over the entire plane R2)
I=Z ZR2
e−(x2+y2)dA =Z∞
−∞ Z∞
−∞
e−(x2+y2)dy dx = lim
a→∞ Z ZDa
e−(x2+y2)dA,
where Dais the disk with radius aand centered at the origin. Show that I=π
(b) An equivalent definition of the improper integral Iin part (a) is
Z ZR2
e−(x2+y2)dA = lim
a→∞ Z ZSa
e−(x2+y2)dA,
where Sais the square with vertices(±a, ±a). Use this to show that
Z∞
−∞
e−x2dx Z∞
−∞
e−y2dy =π
(c) Deduce that Z∞
−∞
e−x2dx =√π
(d) By making the change of variable t=√2x, show that
Z∞
−∞
e−x2/2dx =√2π
(e) Let µand σbe constants. Deduce, from part (d), that
Z∞
−∞
1
√2πσ e−(x−µ)2/2σ2dx = 1
[Remark: This is a fundamental result in probability and statistics]
2. Find the volume of the part of the ball x2+y2+z2≤a2that lies within the cylinder x2+y2=ax
and above the xy-plane.
3. Find the surface area of the part of the sphere x2+y2+z2=a2that lies within the cylinder
x2+y2=ax and above the xy-plane.
4. Use a double integral to find the area of the region inside the circle r= 4 sin θand outside the
circle r= 2.