Why do string instruments need hollow bodies? My textbook states, 'The sound box has a large area, it sets a large volume of air into vibration, the frequency of which is same as that of the string. So due to resonance a loud sound is produced.'
My question is why isn't the air around the string (apart from the hollow body) resonating or resonating sufficiently enough to produce a loud sound without the need for a hollow body? Isn't the hollow body's volume of air negligible compared to 'room/area' where the string is played?
 A: Your textbook is wrong. Stringed instruments such as guitars and violins do not depend on a specific resonance to make them loud. If they did, then they would only be able to play notes within a certain range of frequencies. Resonances exist, and do modify the response as a function of frequency, but they are not essential to the phenomenon.
It is also not true that you need to have a hollow body. Having a hollow body with holes does create a Helmholtz resonance in instruments like violins and guitars, but these instruments would still work just fine without the Helmholtz resonance. Try covering the hole on a guitar and you'll see. The sound is just different.
A: This question has some similarities with that one.
If the strings are fixed in a hard solid object, or in the ground, their vibrations last longer, because less energy is lost to the surroundings.
Because the instrument is built with large thin surfaces, it will vibrate together with the string attached to them. Each string vibrates in a specific frequency, function of its length, material and tension. The instrument is forced to vibrate in the same frequency (forced oscillations).
The string vibrations spread then into membrane vibrations. That surfaces lose more quickly its energy to the air than the strings alone, what increases the power of the sound. On the other side, the hollow chamber avoids that the sound fades too quickly, because the sound waves can reflect back and forth inside it.
A: Yes, the room has a lot of air, but most of it isn't in direct contact with the vibrating string. In order for the string to make much sound, it needs to transfer some of its energy to the surrounding air, but there are a couple of issues which make that difficult.
Firstly, the string has a relatively small surface area, so it simply doesn't directly contact very much air.
Secondly, the string isn't very effective at transmitting its vibrations to the surrounding air. It's a bit like if you get a thin rod and wave it back and forth in water. Yes, it makes a few ripples, but the rod mostly just cuts through the water. But if you get a wide paddle and wave it around, then it's easy to make large waves in the water. Similarly, a vibrating string mostly just cuts through the air, rather than making useful pressure waves in it.
With air, it's even harder to effectively induce waves than in water, because it's so compressible. Obviously, pressure waves do travel through the air, but it's a lot harder to transmit such waves through a gas than through a denser, stiffer medium.
So stringed instruments generally have some kind of soundboard: a surface with a relatively large area that's in contact with a lot of air. The string can efficiently transmit its vibrations to the soundboard, and the soundboard can, in turn, transmit those vibrations to the air.
On a guitar, the soundboard is the front of the body. Here's a diagram, courtesy of Wikipedia:

The soundboard of a piano is inside the body:

(source)
An important concept here is acoustic impedance.

Acoustic impedance and specific acoustic impedance are measures of the opposition that a system presents to the acoustic flow resulting from an acoustic pressure applied to the system. [...] There is a close analogy with electrical impedance, which measures the opposition that a system presents to the electrical flow resulting from an electrical voltage applied to the system.

Unfortunately, that Wikipedia article is heavily technical and mathematical, without much physical detail.
The soundboard of a musical instrument, or the membrane of a loudspeaker, doesn't need to have a lot of back and forth movement. It just needs to vibrate relatively gently. Large movement wastes energy by causing bulk movement of the air, rather than inducing the desired pressure waves.
A: Existing answers have correctly identified that the issue here is not resonance but impedance matching. I would like to write a little to show what we mean by impedance matching.
The issue here is: how to get the energy which is in the vibration of the string to be transferred to the air in the room? The string itself pushes a tiny volume of air near to it, and does not cause much pressure difference even in that air (it just slides past) so basically the string does not transfer energy directly to the air. Rather, the string causes the bridge to vibrate, and the bridge causes the sound-board to vibrate, and the sound-board exerts force on a wide area of air, enough to cause a pressure change, and thus the energy is converted into sound waves in the air. Here the 'bridge' is the name used in instruments such as violins and acoustic guitars, and by 'soundboard' I mean the part of the instrument on which the bridge rests. Note, it is a part which itself is thin enough to vibrate. Electric guitars often do not have a soundboard and consequently do not sound loud unless they are amplified electronically. Inside some instruments a further feature called a post transfers vibrations between the soundboard and the back of the instrument. Pianos have structures filling equivalent roles, but going by different names.
Notice that I have said nothing at all about resonance so far. In fact makers of instruments have to be careful to make sure the sound box does not resonate too much at any particular frequency, or else the instrument would make that note come out too loud compared to all the others. Rather, the resonances of the instrument have to be not too precise; they must merge with one another, at least in the audio region of the spectrum. This is why stringed instruments are made of materials such as wood, rather than steel or glass. Note, I am not saying that resonance plays no role at all. I am just saying that it is not the main issue here if we understand the term in its physics meaning. In physics, a resonance is an oscillation at a natural frequency which can be built up by applying a small driving force at the right frequency over a length of time. But, as I already said, what we require in stringed instruments (as opposed to pipe organs) is that such resonances merge and therefore none of them build up very much.
The technical term "impedance matching" is all about the following concept.
Take a line of steel balls, such as ball bearings. Or you can try to experiment with coins sliding on a flat surface. First suppose the balls are all of the same mass and material. When you line them up so that they are touching, and then roll one of them towards the end of the line, so that it hits, then the impulse is transferred down to the line and the one on the other end moves off.
Now consider the same experiment, but instead of using balls all the same size, take
a line of large ball bearings, and roll a small one towards the end of the line, so that it hits. Again the impulse is transferred down the line. Now you might think that we can compensate for the small mass of the ball we fired by firing it faster. So we could give it the same kinetic energy that we used in the first experiment. But what happens in this second experiment is that the little ball bounces off! It does not manage to transfer much of its energy to the line of larger balls. It comes away with almost the same energy it started with.
The lesson of this example is that transfer of mechanical energy from one thing to another involves other issues as well, such as momentum, and as a result when two things of very different properties interact, you tend to get reflection, whereas when two things of similar properties (e.g. density) interact, you tend to get transmission. The same thing happens with waves on a string. For example if you join a heavy string to a light string then when a wave travels along the light string towards the join, at the join it mostly reflects off and comes back. So the heavy string does not receive the energy.
Now comes our third experiment. Imagine now a line of balls of different masses, arranged along a line in a gradual progression from lightest to heaviest. Now when a light ball hits the end of the line, it will transfer most of its energy and come to rest. Then the next ball transfers most of its energy to the next, and so on all the way down the line. In this case there is good (i.e. almost 100 percent) energy transfer and we
say the impedances are matched.
The comparison with the stringed instrument is that the string and the surrounding air have very different impedance so the energy in the string does not get transferred directly to the air. However the string and the bridge are matched, and the bridge and the soundboard are matched. The result is that a vibration of large amplitude but small area at the string becomes a vibration of small amplitude but large area at the soundboard. And that gets converted to a vibration in the pressure of the air.
Experiment at home
If you have one of those little metal "musical boxes" that make a short tune when you wind a handle, then try the following experiment. First remove any sound box that may be provided, and just hold the metal frame in your hand and turn the handle. You can hear the tune but it is quiet. Now push the metal frame against a hard wooden table top and turn the handle. It is much louder!
A: Resonance is when a wave-like phenomenon happens in a length (string), area (drum head) or volume (sound box) whose geometry reflects those vibrations to constructively interfere. The air around a string will vibrate, but there's nothing to reflect those vibrations so they disperse and decrease in amplitude as they leave the string (as waves mostly do).
A stringed instrument is kind of complicated. First, the strings (the simplest part) are plucked or bowed so that they resonate with a specific frequency based on their geometry (length, mass and tension). Almost instantly, those waves travel (mostly) through the bridge (and likely a sound post) to vibrate the body (top and bottom) of the instrument. As the body of the instrument vibrates, the air inside the body resonates with those vibrations and that (louder) sound escapes to the room through small holes on the body.
Each of these three processes involve resonance and the geometry of the string and the instrument (both body and cavity) are vital to the sound. Viols have curves to resonate at a broad range of frequencies rather than a few specific frequencies as would cuboids and spheroids. The length of full-sized violins and cellos are set by physics to be long enough to vibrate at their lowest note, while the bio-mechanics of the viola and bass render those lengths un-playable.
You are right that there is much more air in the room than the body of most instruments, but the room usually doesn't share a solid connection to the string, and walls aren't usually made of a resonant material.
A: A string vibrating in the air faces a very low impedance; it transfers very little energy to the air. This is why solid-body electric guitars have a lot of sustain.
An acoustic stringed instrument has a sound board. The string is mechanically coupled to some point on the sound board via a bridge. That board moves in response to the string vibrations and its large surface area transfers pressure to the air: the sound board supplies an impedance match between the string vibration and the air.
OK, so why don't we make instruments with just a soundboard and be done? The enclosure around the sound board, with the holes, create a tuned resonator which emphasizes certain frequencies of the sound. The enclosure also prevents cancellation of frequencies.
If you know anything about loudspeakers, you know that an "open back" cabinet has a different sound from a "closed back". An acoustic guitar is much like a closed-back loudspeaker enclosure (and the sound hole is like a tuned port in the front of a speaker). The closed back prevents cancellation of the fundamental mode of the sound-board's vibration. Cancellation occurs when the sound wave coming from the back of the board meets the sound wave from the front of the board which is in opposite phase. Cancellation would result in weaker bass response.
In some cases, a partial enclosure further improves the impedance match to transfer the energy into the air. For instance, your vocal tract has a decent resonant cavity that produces audible sound. Yet, if you put a megaphone into your mouth, a partial enclosure, your voice becomes louder still: the shape of the megaphone allows you to transfer more of your energy into the air through a better impedance match.
A: The string itself does not move the air much, does not produce much sound - hear an electric guitar without the amplifier!
In a violin it is the bridge that couples the movement of the string to the sound board which sets the air in motion.
The hollow body acts like a Helmholtz resonator.
A: The soundboard is the crucial component in acoustical amplification of string vibrations. The strings pass over the bridge, which is solidly attached to some part of the soundboard. A vibration at the bridge sets a good part of the soundboard in motion. This in turn vibrates the air close to the soundboard, and launches sound waves into the air, which we can hear. An important point here is that the "amplification" of the string vibration does not actually add any power that was not already present in the vibrating string. All we have done is transform the vibration from a small, stiff source (the string and bridge) to drive a large, soft destination (the soundboard then the air). This is a bit like using a lever to increase the effect of a force, without actually adding any power.
This soundboard effect is exploited in loudspeakers, where the soundboard is the large, lightweight cone, and the motive force comes from an electric current in a coil, in a magnetic field. The "motor" part of the loudspeaker, the coil in the magnetic field, would flap about ineffectually when driven by an electric signal, were the motor not attached to a relatively large cone, which actually drives the surrounding air.
Most string instruments have an enclosed sound box, the soundboard being one side of this box. This is partly practical in providing a stiff construction for attachment the strings to a neck, bridge, and soundboard. Bear in mind that the strings are under considerable tension, which would pull apart a weak construction.
This box behind the soundboard has acoustical effects as well. One problem with using a vibrating soundboard is that each side of the soundboard vibrates in opposite phase. This would mean that, at some distance away, the air vibrations tend to cancel out. This cancellation is not total. In particular, sound from the far side of the soundboard must "go round a corner", to get to the front and mix with the sound from the front. Low frequencies readily spread around obstructions, whereas high frequencies tend to be deflected. This depends on the size of the obstruction, relative to the wavelength of the sound. One effect putting a closed box behind the soundboard is to obstruct sound from the rear of the soundboard, and prevent cancellation of sound from the front, particularly at low frequencies. This effect is also exploited in almost all low frequency loudspeakers, in order to improve the bass response. Some loudspeakers, such as electrostatic types, rely on having a large soundboard area, but this is not common.
Now we come to the matter of resonance. There is some unfortunate conflict between common usage, and physics or engineering terminology. Informally, an object is "resonant" if it readily makes a sound when set in motion. A thin board of wood or a metal saucepan are resonant in this sense, whereas a bucket of sand or a woolly jumper are not resonant, in the informal sense. But to an engineer, a resonance means a tendency for a system to respond more strongly to vibrations in some narrow range of frequencies, around the resonant frequency. Organ pipes and stretched strings are resonant in this stricter sense. Real acoustic/mechanical systems tend to have many resonant frequencies.
The problem with resonances in string instruments is that we want the soundboard to be readily set in motion to make a sound (informal resonance), but we want this to happen more or less equally for all frequencies that the strings can produce. The latter requirement means that we want to avoid prominent resonant frequencies, that would amplify some notes at the expense of others. In the end, we end up with a system that does have specific resonances, but they are artistically controlled to "sound right", and make the instrument respond to the player. If we try to control unwanted resonances too much, this detracts from the desirable responsiveness of the soundboard amplification.
As a final note on resonances, there is the matter of soundholes. Specific low frequency resonances of the soundboard and the air in the box can be exploited to boost the lower frequency amplification. This gives an instrument a fuller and deeper sound. In loudspeaker technology, this effect is used to produce a bass-reflex cabinet, which has an improved bass response compared to a plain closed box.
Not all string instruments have a sound box to improve the bass sound. The piano has a very large soundboard, which helps to prevent cancellation of bass frequencies. The banjo uses a drum skin as a soundboard, with thin shell rather than an enclosing box. The banjo sound is notably lacking in bass.
A: Short answer:
They don't need hollow bodies, but hollow bodies amplify the sound, while also offering an additional opportunity to colour the sound. That makes them very popular.
Long answer:
A typical instrument works like this:

*

*Something creates noise (noise contains all frequencies)

*The instrument filters the frequencies, resulting in notes of particular timbre.

Filtering happens based on whether the frequency's corresponding wavelength fits a whole number of times in the space it is given. E.g. with a wavelength of 2cm, the frequency needs spaces of 2 or 4 or 6cm, etc.
Filtering happens mainly in two places: the string and the body.

*

*A string of a particular length and tension can only host one frequency (the dominant frequency) and its multiples (harmonics). Everything else is quickly lost after picking the string. That results in a generic and debatably unpleasing sound.

*Using the shape of the body, you can filter out more frequencies, giving more personality to the resulting sound. An interesting point here is that the body doesn't only filter the already filtered sounds of the strings. It also works with some of the initial noise.

Everything I wrote so far applies to both solid and hollow bodies. A hollow body adds two more things:

*

*With more surface, the energy from the vibrating instrument tranfers to the air faster. This results in a louder sound, with reduced longevity. The overall output is increased too, since less energy is lost inside the instrument before being transefered to the air.


*By emitting the sound into a confined space, you get an additional chance to shape the sound: the wavelengths need to fit in the confined space. This is very useful for pronouncing bass. Wavelengths are shorter in air than in solids, like the solid parts of an instrument. This means that a solid-body instrument can't keep its long wavelengths (a.k.a. bass) as well an instrument that is mostly air (by volume).
About me:
I am not a physicist, but I've spent a lot of time around guitars of varying shapes and materials.
A: A sound hole is an opening in the body of a stringed musical instrument, usually the upper sound board. Sound holes have different shapes:
round in flat-top guitars and traditional bowl-back mandolins;
F-holes in instruments from the violin family, archtop mandolins and in archtop guitars;
C-holes in violas da gamba;
rosettes in lutes;
D-holes in bowed lyras.
Some instruments come in more than one style (mandolins may have F-holes, round or oval holes). A round or oval hole or a rosette is usually a single one, under the strings. C-holes, D-holes and F-holes are usually made in pairs placed symmetrically on both sides of the strings. Most hollowbody and semi-hollow electric guitars also have F-holes.
Though sound holes help acoustic instruments project sound more efficiently, sound does not emanate solely from the sound hole. Sound emanates from the surface area of the sounding boards, with sound holes providing an opening into the resonant chamber formed by the body, letting the sounding boards vibrate more freely, and letting vibrating air inside the instrument travel outside the instrument.
Alternate designs :
Some Ovation stringed instruments feature a particularly unique soundhole architecture with multiple smaller soundholes that, being combined with a composite composite bowl back body, are said to produce a clear and bright sound.
Tacoma Guitars has developed a unique "paisley" soundhole placed on the left side of the upper bout of their "Wing Series" guitars. This is a relatively low-stress area that requires less bracing to support the hole.
A few hollowbody or semi-hollow electric guitars, such as the Fender Telecaster Thinline and the Gibson ES-120T, have one f-hole instead of two, usually on the bass side.
B&G Guitars, a private build guitar company from Tel Aviv, Israel, uses their signature "backwards" sound holes on their guitars
Holes not positioned on the top of an acoustic guitar are called soundports. They are usually supplementary to a main soundhole, and are located on an instrument's side facing upward in playing position, allowing players to monitor their own sound.
