COLUMBUS, Ohio — The White House the prize, President Barack Obama and Mitt Romney raced through a final full day of campaigning on Monday through Ohio and other battleground states holding the keys to victory in a tight race. Both promised brighter days ahead for a nation still struggling with a sluggish economy and high joblessness.

“Our work is not done yet,” Obama told a cheering crowd of nearly 20,000 in chilly Madison, Wis., imploring his audience to give him another four years.

Romney projected optimism as he neared the end of his six-year quest for the presidency. “If you believe we can do better. If you believe America should be on a better course. If you’re tired of being tired ... then I ask you to vote for real change,” he said in a Virginia suburb of the nation’s capital. With many of the late polls in key states tilting slightly against him, he decided to campaign on Election Day in Ohio and Pennsylvania, where he and Republicans made a big, late push.

The presidency aside, there are 33 Senate seats on the ballot today, and according to one Republican official, a growing sense of resignation among his party’s rank and file that Democrats will hold their majority.

The situation was reversed in the House, where Democrats made no claims they were on the verge of victory in pursuit of the 25 seats they need to gain control.

National opinion polls in the presidential race made the popular vote a virtual tie.

In state-by-state surveys, it appeared Obama held small advantages in Nevada, Ohio, Iowa and Wisconsin — enough to deliver a second term if they endured, but not so significant that they could withstand an Election Day surge by Romney supporters. Both men appealed to an ever smaller universe of undecided voters.

More than 30 million absentee or early ballots have been cast, including in excess of 3 million in Florida. The state also had a legal controversy, in the form of a Democratic lawsuit seeking an extension of time for pre-Election Day voting.

There were other concerns, logistical rather than legal.

Officials in one part of New Jersey delivered voting equipment to emergency shelters so voters displaced by superstorm Sandy last week could cast ballots. New York City made arrangements for shuttle buses to provide transportation for some in hard-hit areas unable to reach their polling places.

Judging from the long early voting lines in some places and the comments made in others, the voters were more than ready to have their say.

“I watch the news all the time, and I am ready for it to be over,” said Jennifer Walker, 38, of Columbus, Ohio, who said she took time off from work to attend the president’s speech during the day in a show of support. “I feel like he is getting better with the economy. I don’t think it’s hopeless. It takes time.”

But Bryan Dobes, 21, a University of Iowa student from suburban Chicago, voted for Romney on Monday and said unemployment and spending have been too high under Obama. “He promised a lot of hope and change, and I’m not seeing it,” he said of the president.

“No retreat, no surrender,” sang rock icon Bruce Springsteen, warming up Obama’s crowd on a frosty morning outside the State Capitol in Madison, Wis. The Boss then boarded Air Force One for his first flight. “Pretty cool,” he judged it.

Romney had Kid Rock and the Marshall Tucker Band in the wings for his late appearances in Ohio and New Hampshire.

“This is it,” the challenger said in a last-minute emailed request for campaign donations.

“I will lead us out of this economic crisis by implementing pro-growth policies that will create 12 million new jobs. With your help, I will deliver real change and a real recovery. America will be strong again.”

In his longest campaign day, Romney raced from Florida to a pair of speeches in Virginia to Ohio and then an election eve rally in New Hampshire.

Obama selected Wisconsin, Ohio and Iowa for his final campaign day, an itinerary that reflected his campaign’s decision to try and erect a Midwestern firewall against Romney’s challenge.

Vice President Joe Biden and Republican running mate Paul Ryan of Wisconsin went through their final campaign paces, as well.

In Sterling, Va., not far from Washington, the vice president accused Republicans of running away from their record, but added, “a leopard can’t change his spots.”

Ryan started out in Reno, Nev., where he said the president has come up short in his promises to change Washington and repair the economy.

“This may be the best that Barack Obama can offer, but this is not the best America can,” he said.

before flying off to Colorado and Ohio. Then it was home to Wisconsin, where he is on the ballot for re-election to Congress in case Republicans were unsuccessful in the presidential campaign.

Conscientious to the end, supporters kept knocking on doors in search of a possibly decisive vote.

In Enfield, N.H., Obama volunteer Sarah Ayres recalled driving up a deserted dirt road, unsure if she would find the house she was looking for. She turned down a long driveway, she said, got out of her car, and was met by a friendly, white goat.

“There were no people home, but the goat was there, so I don’t know if I should count that as a contact,” she joked.