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The illuminated area is showing where daytime was at the moment of this past full moon, 15:47 UTC, Dec. 3rd, while the darkened area shows where it was night at that time.

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Worldwide map via the US Naval Observatory.

Day and night sides of Earth at the instant of full moon

(2017 December 3 at 15:47 UTC).

The shadow line at left depicts sunrise December 3

and the the shadow line at right represents sunset December 3.

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Why is the illuminated area shaped like a bell-curve? Does the sun shine all the time with this pattern of light?

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The caption refers to the "shadow line at left" and "the shadow line at right," What about the shadow line at the top and the shadow line at the bottom?

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Furthermore, why does the shadow line curve in between the left and the top, and again between the right and the bottom?

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From the very informative website:
http://earthsky.org/tonight/december-cold-moon-tonight
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Tonight – December 2, 2017 – watch for the nearly full moon. As seen from around the world, the moon will shine pretty much all night long, starting around sunset on this night. Although the calendar gives December 3 as the full moon date, the exact clock time (and possibly the date) of the full moon varies by time zone. No matter where you live worldwide, the moon will appear plenty full to the eye both tonight and tomorrow night.
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By the way, this full moon will be the only full moon supermoon to light up our sky in 2017. What’s a supermoon? Read more about it in our post for December 3, 2017.
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The December 2 moon shines in front of the constellation Taurus the Bull. Despite the lunar glare, you still might be able to make out Aldebaran, Taurus’ brightest star, and possibly the Pleiades star cluster.
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Read more: Occultation of Aldebaran on the night of December 2-3, 2017
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Worldwide map via the US Naval Observatory. Day and night sides of Earth at the instant of full moon (2017 December 3 at 15:47 UTC). The shadow line at left depicts sunrise December 3 and the the shadow line at right represents sunset December 3.
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The full moon, our nocturnal sun, stays out throughout the night, and then sleeps in during the day. The December full moon, like the June sun, climbs up high as seen from the Northern Hemisphere sky. A full moon near the winter solstice travels a high path across the sky and stays in the sky for all hours of the night. That’s why one of the names for this full moon is the Long Night Moon.
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From the Southern Hemisphere, where the days are long and the nights are short, the December full moon follows the low path of the winter sun. After all, it’s the hot season in that hemisphere now.
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From the contiguous United States, the moon reaches the crest of its full phase during the daylight hours on December 3, when the moon is beneath our horizon. At North American time zones, the moon turns precisely full – resides 180^o from the sun in ecliptic longitude – at 11:47 a.m. AST, 10:47 a.m. EST, 9:47 a.m. CST, 8:47 a.m. MST and 7:47 a.m. PST.
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However, if you live in Alaska or Hawaii, the moon turns full before sunrise December 3, meaning the moon will be in your sky at the instant of full moon, which takes place on December 3, 2017, at 15:47 Universal Time (UTC) – or 6:47 a.m. AKST (Alaska Standard Time) and 5:47 a.m. Hawaii-Aleution Standard Time (HAST).
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Bottom line: From around the world, the December 2, 2017 moon shines in the east at nightfall, climbs highest for the night around midnight and sets in the west around sunrise December 3. For the contiguous United States, the moon will turn precisely full during the daylight hours on December 3, when the moon is below the horizon.
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Speaking of the Naval Observatory, the North Koreans would love to know that some Americans are flat-earthers - they would likely hope that some of the people responsible for ICBM tracking are ignorant of the earth's curvature, so the Korean missiles would then have a decided advantage.
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Yes, it's an artist's depiction, that is, a graphic artist. The curving line is plotted by observational data.
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It's not "what they think is occurring" it's what has been tabulated and recorded, in fact.
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Remember the total solar eclipse, how all the data was calculated and predicted and the eclipse did precisely what was anticipated?
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The scientists who foretold where the moon's shadow would fall minute by minute were not flat-earthers BTW.
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It depicts a snapshot of where the sun's daylight illumines the earth and where it does not, at one moment in time.
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There is no way to get a photograph of this because it's shown on a Mercator projection, which is artificial, and necessarily so because to make a spherical object appear flat for illustration you have to include particular distortions.
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But the distortions don't have to interfere with the overall shape of the day/night transition profile.
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The curve shows you where daytime changes into night time.
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Every day, this same pattern appears on the face of the earth. Every day. Every day. Every day.
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So it's not a fluke, and it's not pretend, and it's not a fantasy. It is reality.
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If you can explain how to get that curve when you think the sun is whirling around like a dervish over your "flat" earth, be my guest.
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And how is this pattern supposed to prove a Globe Earth Neil?