Parse out exon coordinates from bed file for each gene
There are multiple ways to go about it. On the command line you can make a 1 line BED file:

chr1   11868   12227


And then bedtools intersect with it.

In R, you could load your original BED file and use GenomicRanges:

library("GenomicRanges")
bed = read.delim("foo.bed", header=F)  # Rename this
# N.B., BED files use 0-based coordinates, I've switched to 1-based here
exons = GRanges(seqnames=levels(bed$V1),
                ranges=IRanges(start=bed$V2+1, end=bed$V3),
                strand=bed$V4)
# Again, I switched to 1-based coordinates...you don't have to do that
region = GRanges(seqnames="chr1",
                 ranges=IRanges(start=11869, end=12227),
                 strand="+")
theExonsYouWant = subsetByOverlaps(exons, region, type="within")


You'll need to either specify a strand in your region or use ignore.strand=T in subsetByOverlaps(). I hope you don't have any genes that overlap on the same strand, since they'll prove problematic.