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There are many applications for the area between curves. For example if the curves represent the speeds,then the area is the difference between the total distance traveled.

Another application is for two investments with different interest rates and the area represents the difference between the total interest gained by accounts.

For instance, computing that area of the disk$${(x,y)inmathbb{R}^2,|,x^2+y^2leqslant r^2}$$ is the same thing as computing the areas between the curves$$begin{array}{ccc}[-r,r]&longrightarrow&mathbb{R}^2\x&mapsto&sqrt{r^2-x^2}end{array}$$and$$begin{array}{ccc}[-r,r]&longrightarrow&mathbb{R}^2\x&mapsto&-sqrt{r^2-x^2}end{array}.$$So, unless you find the problem of computing the area of a disk beneath you, you can see that it is as interesting as the problem of finding the area between two specific curves.

There are loads of examples in the real world outside of pure mathematics.

In terms of materials, finding the area under the stress-strain curve tells that material's toughness - how much abuse (energy) it can take (absorb) before it begins to fracture. If you take the difference between two materials' toughness, you can see which one is tougher, which is something that's not always apparent at just looking at their graphs.

Example: Say we have materials A and B, each with stress-strain curves given by $sigma(epsilon)$.

begin{cases}
begin{align}
sigma_A(epsilon) &= 5 epsilon, &epsilon in (0,1)\
sigma_B(epsilon) &= tanhepsilon, &epsilon in (0,3)
end{align}
end{cases}

These functions, by the way, are wildly unrealistic in most occurrences, but the concept is still relevant.

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It's not exactly clear which material has the higher toughness. Material A's looks like a lot more, but maybe the longevity of material B makes up for its low yield. We could find the area under each and see which is higher but we could assume material A is tougher and find the area between A and B.

$$Delta U = int_0^3 Big( 5epsilon big(1 - h(epsilon - 1)big) - tanh epsilon Big) , depsilon =frac{5}{2} - logcosh3approx 0.19.$$

So material A is actually hardly more tough than material B.

In terms of systems, you can design an integral-feedback control system that reduces the error in the output from the desired output. In practice, these are PID controllers (Proportional, Integral, Derivative), all 3 of which are focused on changing the actual output to what it should be based on the desired output. The goal is for the error to be zero, i.e. in the integral controller, you're looking for the area between the desired output and actual output to be exactly zero.

For instance, suppose you're driving and you click on the cruise control. You've just told the car you want it to keep going the speed that you are going right this moment. Now the road starts leading you uphill. The car is going to slow down initially, but it is also going to do everything it can to keep you going how fast you told it to go. That is, going uphill, it has encountered some error between the actual speed and desired speed and it wants to bring that error back to zero. The same applies for going downhill as well.

Another example could be controlling the temperature inside an oven, perhaps an expensive one with variable heating elements using a similarly styled controller.

These are just a few examples; there's really no shortage of applications.

Economics.

Example 1. The area between the demand (supply) function and the market-clearing (equilibrium) price shows the consumers (producers) surplus:

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Source: Wikipedia.

Note that the consumers are willing to pay a price for a good up to their demand function, but they pay the equilibrium price and save the difference (red area).

Example 2. The area between the marginal revenue and the marginal cost functions shows the net profit (loss).

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Source: Slide $13$ at Slideshare.

Another aspect worth pointing out is that definite integrals solve many problems other than finding the area between curves- e.g. finding total net change in some system given the instantaneous rates of change. Finding the area between curves is an application that happens to be particularly easy to visualize.

Given two interest rate functions of time (the instantaneous rates of change of the values of two investments), the integral of the difference between them with respect to time is the difference in the returns of the two investments they represent over the interval of integration.

Chromatographic peak assignment associates mass with the area between a sensor response curve and the baseline curve. A few examples are shown here. (The Wikipedia has this

although the baseline is not explicitly shown. That determining the baseline between, say, 20 and 25 minutes doesn't yield something simple should be easy to see.)

Integration is the art of adding together very many, very small things. There are many things other than areas which can be added together this way, but areas are nice to represent visually, and relatively easy to intuit about (as long as your examples aren't too pathological). So that's the analogy most authors and lecturers go for when they introduce integration.

Note that once the things you add become something different than real numbers (like vectors, or complex numbers), then the area analogy breaks down (it even struggles somewhat for negative numbers). For someone who cannot think of integration as anything other than finding areas, that's going to be a difficult transition. So I definitely would advise you to think about integration in other terms as well.

One standard example would be if you had velocity as a function of time. You could draw the graph of that function and say that the area under the graph represents distance travelled. But this becomes very difficult to generalize once, say, you add direction of travel into the mix (i.e. you're traveling in a plane rather than on a line). You can somewhat still recover the intuition for total distance travelled, but for net distance (i.e. the straight line distance from starting point to end point) it becomes a hassle.

On the other hand, one could forgo the area analogy completely and just say that for a very small time interval, you travelled a very small distance, and then add all those small distances together to find the total distance travelled. This is much easier to intuitively generalize from travel along a line to travel in a plane (especially if you're interested in net distance travelled), and on to higher dimensions.