SQL Server 2016

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Declare @Request table (Employee varchar(50),start_date date,end_date date)
Insert Into @Request values
 ('Jane Smith','2018-08-10','2018-08-14')
,('Tom Jones' ,'2018-08-12','2018-08-18')
,('Some Other','2018-08-22','2018-09-03')

Select * into #Request from @Request

Declare @D1 date = '2018-08-01'
Declare @D2 date = '2018-08-31'

Declare @Cols varchar(max) = (Select Stuff((Select ',[' +cast(N as varchar(25))+']' From (Select Top (DateDiff(DAY,@D1,@D2)+1) N=Row_Number() Over (Order By (Select Null)) From  master..spt_values n1) A For XML Path ('')),1,1,'') )
Declare @SQL varchar(max) = '
Declare @D1 date = '''+cast(@D1 as varchar(50))+'''
Declare @D2 date = '''+cast(@D2 as varchar(50))+'''

Select *
 From  ( 
		Select A.Employee
			  ,Item      = day(d)
			  ,Value     = case when D between start_date and end_date then ''R'' else '''' end
		 From #Request A
		 Cross Join (
						Select Top (DateDiff(DAY,@D1,@D2)+1) D=DateAdd(DAY,-1+Row_Number() Over (Order By (Select Null)),@D1) From  master..spt_values n1
					) B
       ) src
 Pivot (max(value) for Item in ('+@Cols+') ) pvt
'
Exec(@SQL)

Employee	10	11	12	13	14	15	16	17	18	22	23	24	25	26	27	28	29	30	31
Jane Smith	R	R	R	R	R
Some Other										R	R	R	R	R	R	R	R	R	R
Tom Jones			R	R	R	R	R	R	R