Math495 Assignment: Analysis exercise
Ngoc Chi Le 403578789
November 30, 2007
Problem 1. Let Ω ⊂R3be bounded set, measurable and f: Ω →Rbe
bounded function in Ω. Show that fis integrable in Ω if and oly if for all
> 0 there exists partition ∆ of Ω such that:
S+(f, ∆) −S−(f , ∆) < ,
where S+(f, ∆), S−(f , ∆) is upper Darboux’s sum and lower Darboux’sum.
Problem 2. Find the volume of object bounded by following surfaces:
x2+y2+z2=a2;x2+y2+z2=b2
x2+y2=z2;z≥0 (b > a > 0).
Problem 3. Compute the integrand
IC
(−y2+eex)dx + arctgydy,
where Cis the boundary of the domain Gbounded by y=x2;x=y2,
counterclockwise.
Problem 4. Let f:Rk→Rbe continuous and α∈R. Show that the set
A(α) = {x∈Rk:f(x) = α}is closed set.
Problem 5. Show that ∞
Z
−∞
e−x2dx =√π
1
