How does modulation work in simple language?

I am reading modulation from this site (am a beginner). In the end, they've given this example: I am having trouble understanding how exactly they arrived at the two modulated results for AM and FM. I have attempted to understand it, and this is my attempt (I am assuming the $x$-axis is time axis):

1. Amplitude Modulation: At each time $t=t_0$, they have taken an increase factor as $k=\frac{\text{amplitude of modulating wave at } t_0}{\text{constant amplitude of carrier wave}}$. Hence, to obtain the modulated result at each instant, they've done $\text{Final displacement}=\text{Original displacement of carrier wave}\times(1+k)$. Negative amplitude of modulating wave implies decrease of carrier wave's displacement, and positive amplitude of modulating wave implies increase.
2. Frequency modulation: I'd say a similar increase factor is at work here, but instead, that factor is now affecting the frequency of the carrier wave instead of its amplitude.

I am sorry if these are beginner knowledge but I don't have any good book to study from. I checked Wikipedia for these two as well. My guess for amplitude modulation's working seems reasonably close to their mathematics, especially the $y(t)=[1+m(t)]\cdot c(t)$. But, I could not understand why an integral is employed in frequency modulation. What is it supposed to do?

I hope I have a reasonably detailed and specific question here. I am looking for simple language explanations. Thank you!

• @StanleyPawlukiewicz Thanks for your remark. Did you mean to link to the "Wide-band FM radio" section on that Wikipedia page? I read it and it mostly seems to be concerned with the "Armstrong slowly developed what eventually resulted in wide-band FM"..."Armstrong had a standing agreement to give RCA "..."However RCA, which was heavily invested in perfecting television broadcasting, chose not to invest in FM"..."Armstrong gave a formal presentation of his new system". Basically, all the "meta" aspects of its development are covered, so I am not sure how that answered my question. Feb 18 '18 at 4:48
• @StanleyPawlukiewicz Thanks for your 2nd remark, but I still do not see how that helps me plot the graph for a resultant modulation given a carrier wave and the modulating wave. Sorry but I am a beginner. Thank you! Feb 18 '18 at 4:50

In order to understand the integral occurring in frequency modulation you need to understand the relation between phase and frequency. A sinusoid with frequency $f$ can be written as

$$x(t)=\sin(2\pi f t)\tag{1}$$

where the argument $2\pi f t$ is called the phase $\phi(t)$. The frequency is the derivative of the phase divided by $2\pi$:

$$f=\frac{1}{2\pi}\frac{d\phi(t)}{dt}\tag{2}$$

If the frequency is not constant, the relation $(2)$ still holds. Assume now that the frequency is a function of time $f(t)$. From ($2$), the phase can be determined from the frequency by integration:

$$\phi(t)=2\pi\int_{0}^tf(\tau)d\tau\tag{3}$$

In frequency modulation, the frequency of the carrier is changed according to the message signal $m(t)$ around a given (average) carrier frequency $f_c$:

$$f(t)=f_c+\Delta f\cdot m(t)\tag{4}$$

where $\Delta f$ is the maximum frequency deviation, assuming that $|m(t)|\le 1$. From $(3)$ and $(4)$ we can obtain the phase

$$\phi(t)=2\pi f_ct+2\pi\Delta f\int_0^tm(\tau)d\tau\tag{5}$$

From $(5)$, the frequency modulated signal can be written as

$$s(t)=\sin(\phi(t))=\sin\left(2\pi f_ct+2\pi\Delta f\int_0^tm(\tau)d\tau\right)\tag{6}$$

Your understanding of amplitude modulation seems to be correct. You simply multiply the carrier with a non-negative signal that is obtained from the message signal plus a constant. The constant must be chosen such that the sum remains positive.