Q: I am having trouble with my iPad 2 screen shutting down ages when I am on Web.  Could this be a battery problem?    Or Wifi?

If you have it connected to the charger does this happen?

I see you have an iPad 1. Could be the battery needs replacement. See link below.

Try this first - Reset the iPad by holding down on the Sleep and Home buttons at the same time for about 10-15 seconds until the Apple Logo appears - ignore the red slider - let go of the buttons. (This is equivalent to rebooting your computer.)

The quickest way (and really the only way) to charge your iPad is with the included 10W or 12W (5W on Mini) USB Power Adapter. iPad will also charge, although more slowly, when attached to a computer with a high-power USB port (many recent Mac computers) or with an iPhone Power Adapter (5W). When attached to a computer via a standard USB port (2.5W, most PCs or older Mac computers) iPad will charge very slowly (but iPad indicates not charging). Make sure your computer is on while charging iPad via USB. If iPad is connected to a computer that’s turned off or is in sleep or standby mode, the iPad battery will continue to drain.

Apple recommends that once a month you let the iPad fully discharge & then recharge to 100%.

How to Calibrate Your Mac, iPhone, or iPad Battery

http://www.macblend.com/how-to-calibrate-your-mac-iphone-or-ipad-battery/

At this link http://www.tomshardware.com/reviews/galaxy-tab-android-tablet,3014-11.html , tests show that the iPad 2 battery (25 watt-hours) will charge to 90% in 3 hours 1 minute. It will charge to 100% in 4 hours 2 minutes. The new iPad has a larger capacity battery (42 watt-hours), so using the 10W charger will obviously take longer. If you are using your iPad while charging, it will take even longer. It's best to turn your new iPad OFF and charge over night. Also look at The iPad's charging challenge explained http://www.macworld.com/article/1150356/ipadcharging.html

Also, if you have a 3rd generation iPad, look at

Apple: iPad Battery Nothing to Get Charged Up About

http://allthingsd.com/20120327/apple-ipad-battery-nothing-to-get-charged-up-abou t/

Apple Explains New iPad's Continued Charging Beyond 100% Battery Level

http://www.macrumors.com/2012/03/27/apple-explains-new-ipads-continued-charging- beyond-100-battery-level/

New iPad Takes Much Longer to Charge Than iPad 2

http://www.iphonehacks.com/2012/03/new-ipad-takes-much-longer-to-charge-than-ipa d-2.html

Apple Batteries - iPad http://www.apple.com/batteries/ipad.html

iPhone: Hardware troubleshooting (Power/Battery section also applies to iPad)

http://support.apple.com/kb/TS2802

Extend iPad Battery Life (Look at pjl123 comment)

https://discussions.apple.com/thread/3921324?tstart=30

iOS 7 Battery Life Draining Too Fast? It’s Easy to Fix

http://osxdaily.com/2013/09/19/ios-7-battery-life-fix/

New iPad Slow to Recharge, Barely Charges During Use

http://www.pcworld.com/article/252326/new_ipad_slow_to_recharge_barely_charges_d uring_use.html

iPad: Charging the battery

http://support.apple.com/kb/HT4060

Best Practices for iPad Battery Charging

http://www.ilounge.com/index.php/articles/comments/best-practices-for-ipad-batte ry-charging/

Tips About Charging for New iPad 3

http://goodscool-electronics.blogspot.com/2012/04/tips-about-charging-for-new-ip ad-3.html

How to Save and Prolong the battery life of your new ipad

https://discussions.apple.com/thread/4480944?tstart=0

Prolong battery lifespan for iPad / iPad 2 / iPad 3: charging tips

http://thehowto.wikidot.com/prolong-battery-lifespan-for-ipad

iPhone, iPod, Using the iPad Charger

http://support.apple.com/kb/HT4327

Install and use Battery Doctor HD

http://itunes.apple.com/tw/app/battery-doctor-hd/id459702901?mt=8

To Extend a Device’s Battery Life, Get to Know It Better

http://tinyurl.com/b67c7xz

iPad Battery Replacement

http://www.apple.com/batteries/replacements.html

In rare instances when using the Camera Connection Kit, you may notice that iPad does not charge after using the Camera Connection Kit. Disconnecting and reconnecting the iPad from the charger will resolve this issue.

 Cheers, Tom

What do you mean by "iPad screen shutting down"?  Is the physical screen turning off while browsing the web?  Is the device rebooting (shows an Apple logo after a few seconds as if you just turned it on)?  Or is Safari crashing and returning you to the home screen?