In Powershell 3.0 and above there is both a Invoke-WebRequest and Invoke-RestMethod. Curl is actually an alias of Invoke-WebRequest in PoSH. I think using native Powershell would be much more appropriate than curl, but it's up to you :).
Invoke-WebRequest MSDN docs are here: https://technet.microsoft.com/en-us/library/hh849901.aspx?f=255&MSPPError=-2147217396
Invoke-RestMethod MSDN docs are here: https://technet.microsoft.com/en-us/library/hh849971.aspx?f=255&MSPPError=-2147217396