What you're asking for is doable, but how well it works would very much depend on the amount of data in your table.
You can use a DAX measure to calculate the sum of the duration for any events that fall on a given date. You can do this using a disconnected date table that you slice on (the October 2016 Power BI Desktop release includes a built-in data slicer that allows you to easily pick a range). More information on that here: https://powerbi.microsoft.com/en-us/blog/power-bi-desktop-october-feature-summary/#reportView.
You would then have a CALCULATE measure that SUMS the duration, with FILTER to ensure the event in question lands within the date range selected by the slicer. There's more specifics on this part of the question here: Optimizing Dax & model for "where date between" type queries
However, you're going one step further, in that you don't want to SUM the full duration of events that fall within a given date range - you only want to sum the duration that falls within the given date range.
In order to do that, you have to calculate the duration for each individual row at run time based on the selected date range. You can do this with a measure that uses SUMX (see below), but over a large number of records (thousands, millions) the calculation will start slowing down.
For example, if you have a disconnected date table called Date, and your event table is called Event, you can have a measure such as:
Filtered Duration =
CALCULATE (
SUMX (
Event,
DATEDIFF (
MAX ( MIN ( 'Date'[Date] ), Event[StartDateTime] ),
MIN ( MAX ( 'Date'[Date] ), Event[EndDateTime] ),
SECOND
)
),
FILTER (
'Event',
'Event'[StartDateTime] <= MAX ( 'Date'[Date] )
&& 'Event'[EndDateTime] >= MIN ( 'Date'[Date] )
)
)
MIN('Date'[Date]) in this case formula corresponds to the earliest date in the disconnected date table that is within the selected date range. MAX('Date'[Date]) corresponds to the latest date.
The last part (the FILTER) is saying "only look at events that land on a date within the range selected". The SUMX is saying "for each row, do a DATEDIFF". The MAX within the DATEDIFF is saying "choose the later of either the first date from the slicer, or the start date of the event". The reason for this is that if you select the 15th through the 20th on your date slicer, and an event starts on the 18th, you want to count from the 18th. But if the event started on the 11th, you'd want to count from the 15th. The MIN is doing the opposite with the end date.
If an event falls within the time range entirely, then it will calculate the seconds from start to end. If the end of the event is after the selected time range (for example), then it will count the seconds from the exact start of the event to midnight of the selected end date.
Note that because I've used a date table you wouldn't be able to pick partial days as a date range. You could extend this to include a time table but it gets more complicated (you'd need a separate start & end time table and then account for that logic in the above, already quite complex, formula)
