Quote: "why would I want to know the area under a curve?"
That's really just a generalized case that's used when learning Calculus. In the "real world", your curve might represent some physical attribute. Let's say, every point on the curve represents the speed you're traveling at some point in time. Well, then the area under some segment of the curve represents the total distance you've traveled during that interval.
So if you kept your eye on your speedometer while driving, you could use the area under the curve to calculate exactly how far you've traveled after a certain time.
Quote: "One good use for calculus is finding ideal results."
This right here is an example where calculus is extremely useful.
Say, for example, in something like Economics. You might be able to work out formulas that represent a "productivity", based on a number of variables such as how much a specific task costs, how much of the budget you can allocate to the task, time taken for the task, etc. Obviously you want to work those variables so that you maximize your relative productivity, so you can use Calculus to solve for this and figure out which combination of variables will give you the highest productivity.
Lagrange Multipliers are pretty interesting to read up on for something like this.
Other obvious uses include Physics, as stated. Say you want to know how much fuel to add to a rocket so that it can reach a certain velocity before running out, while taking gravity into account - Well, you would need Calculus to derive the formula for that if you didn't already know it.
In fact, Calculus is used for a lot of proofs and derivations of other formulas. Most of the formulas we use we don't really think about, it's just a matter of plugging in values and getting a result. So you never really "see" the Calculus. But if you look into the proofs of a lot of these formulas, you'll see exactly where it came into play.
