Unemployment in the District of Columbia rose one-tenth of a percentage point, to 10.6 percent, in April, while joblessness in the entire metropolitan area--which includes Fairfax, Arlington, Montgomery and Prince George's counties--declined from 6.1 percent to 5.9 percent.

There were 32,700 unemployed District residents in April, about 100 more than were out of work in March. The number of District jobs increased to 602,000, meaning there were 1,100 more job openings in April than a month earlier.

Summer is coming, accompanied by thousands of high school and college students looking for work. Matthew F. Shannon, acting director of the District's Department of Employment Services, said the student job-seekers will increase local unemployment rolls dramatically.

Aggravating the problem is the lack of jobs for youths in the District's private sector. The Metropolitan Washington Board of Trade and the National Alliance of Businessmen say they expect to find only 300 jobs for Washington teen-agers this summer. Mayor Marion Barry says the District is trying to get another 700 jobs for young people from private employers.

About 17,000 public sector jobs--in city government and nonprofit organizations--are expected to be available for District youths this summer. But about 8,500 of those are funded by the federal government through the Comprehensive Employment and Training Act program, and, as a result, have income restrictions. About 7,500 of the pub lic-sector jobs are funded by the city.

The city-funded jobs have no in- come restriction.