Fiber optic cables are less expensive and require less power to operate. http://computer.howstuffworks.com/fiber-optic4.htm

Most transoceanic transmission is done using fiber cables laid on the seabed. Using satellites would increase latency by a lot.

Here's my stab:

Fiber Optics - wiki
more on Fiber Optics
Satellite Link
Good Quesitons

Not only are fiber optic cables getting cheaper, the transmission of data by way of light/Electro-Magetic Waves(EM) is faster than conventional copper/electrons.

Technicals
"At the beginning of a fiber, the transmitter converts any electrical signal, whether it is analog or digital, into an optical signal. To transmit analog signals, the electronic inputs of the transmitter are used to modulate the light source, while for transmission of digital signals, the digital pulses are converted to light pulses. A converter on the receiver side of the fiber changes the light waves back into electrical signals. Different signals can be transmitted simultaneously using different wavelengths of the light in the optical fiber."

The 'integrity' of data over long transmission depends on the signal strength, which decreases as it traverses the fiber cable (light scattering). To energize the signal modulators are placed along the fiber cable that amplify the signal so that the signal maintains its 'integrity'.

Satellites act like middle-men. They pass-on information that is being requested. There is also the problem with latency associated with satellites. Lets just say day traders DO NOT use a satellite connection but are heavily invested in fiber optics.

Bits traverse oceans via cables Fiber, Geography of Transport System and more recenlty UK to Japan.

This is really a huge topic but fun nonetheless. Learning about this stuff takes a while but connecting all the dots on how data/bits move around is amazing. Humans are interesting beings for sure.

Keep learning.

UPDATE
Interactive Transocean Cable Map
Check-out #11

Does anyone know why fiber optic cables are usually used?

Fiber optics are typically deployed for long haul applications. Copper lines are what have previously been installed in many neighborhoods so many links between our homes and the first hop are copper ("last mile").

Is it just a cost effective choice?

It is expensive to install any link that is subterranean. That is why most of us can not get fiber to our homes. The telcos already paid to get copper to our homes and have little incentive to lay fiber.

How is the integrity of the bits kept when traversing through many routers, even via satellite?

In ethernet (as well as other protocols) data integrity is verified by the use of CRC for layer 2 packets. And checksums for layer 3,4. Other layers may have their own integrity checks.

In TCP (layer 4), when a CRC or checksum fails, often it is required to do a retransmission of the corrupted packet.

How do bits traverse oceans? (I'm assuming by satellite)

There are satellite links. There are also underwater links:
http://ansonalex.com/wp-content/uploads/2011/02/underwater-internet-cable-map.jpg