You must begin with the partition unmounted. If you can't unmount it (e.g. it's your root partition or something else the system needs to run), use something like System Rescue CD instead.
Run parted, or gparted if you prefer a GUI, and resize the partition to use the extra space. I prefer gparted as it gives you a nice graphical representation, very similar to the one you've drawn in your question.
resize2fs /dev/whatever
e2fsck /dev/whatever (just to find out whether you are on the safe side)
Remount your partition.
While I've never seen this fail, do back up your data first!
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The resize2fs man page says: If the filesystem is mounted, it can be used to expand the size of the mounted filesystem, assuming the kernel supports on-line resizing. (As of this writing, the Linux 2.6 kernel supports on-line resize for filesystems mounted using ext3 and ext4.). However, resizing a mounted filesystem is a more dangerous operation, since the kernel could easily freeze or crash while running rarely exercised code, leaving your filesystem in a bad state. – 200_success May 21 '13 at 15:12
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True -- but you don't want to be messing around with your partition table with the filesystem mounted. That's why I started with the fs unmounted. – Flup May 21 '13 at 15:15
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For ext4, might be resize4fs instead on RHEL 5. – Zac Thompson Sep 4 '14 at 20:04
not sure what virtualization you are using but in most cases shouldn't need to shutdown the guest to add/remove the disk
unmount, comment out fstab entry, remove the drive at the hypervisor level then add a new one, check fdisk as to the device name, partition, mount, modify fstab (if the name is the same, just uncomment)
alternatively, you can expand the virtual disk instead of removing and creating a new one and should be able to do hot (without unmounting or changing fstab)