Study help ASAP with a few problems (Laplace, areas, an integral)
1) Let W be the region bounded by the curve y=e^(-x^2) and the x-axis, x is greater or equal to 0. Show that W has finite area when A = .5 sqrt(pi). Then calculate the volume generated by revolving W about the y-axis.
2) f(x)=1/(sqrt(2pi)) * integral from negative infinity to x of: e^[(-t^2)/2] dt
Prove that this integral on the right converse for all real x
3) The Laplace transform of f is the function F defined by setting:
F(s) = integral from 0 to infinity of: e^(-sx) f(x)dx
The domain of F is the set of numbers s for which the improper integral converges. Find the Laplace transform F of f(x)=1 specifying the domain of F.
Thanks for the help!