Area Under Curve: Calculus Senior Year | Uses, Benefits

[Periapsis]

I enrolled in Calculus for my senior year in high school, so far loving it. Anyways, I have been reading ahead and figuring some things out, but on the topic of Integration, what can you use the area under the curve for? I've tried searching around in my textbook, and maybe just my google skills fail, but I can't seem to find a useable situation for the area under the curve?

 

[phinds]

Don't get hung up on the geometric aspect of integration. That's like saying the only thing differentiation is good for is to figure out the slope of a curve.

Differentiation and integration have uses FAR beyond those simple geometric interpretations.

 

[Hyrum]

If you go far enough ahead you'll come across the Fundamental Theorem of Calculus which states an intimate relationship between Integration and Differentiation. It basically says that given a function f(x) for which an antiderivative F(x) is known, then
$$ \int_a^b f(x) = F(b) - F(a)$$

A basic example in physics with real-world application that beautifully illustrates this relationship and shows a need for integration is as follows: An object tracks it's position and records it's total distance traveled at a given time to function d(t). By differentiating this with respect to t, you get it's velocity at any given time v(t)=d'(t). Now, say for some reason you only know the velocity function and need to get distance traveled between two times. You simply Integrate v(t) between those 2 points with respect to t and you have the change in distance.

 

[Periapsis]

Thank you very much! I'll have to read into that Theorem

 

[Hyrum]

You're welcome! Glad I could help :)

 

[HallsofIvy]

I might point out that finding the area under the curve in a way that was very similar to "integration" goes back to Archimedes (though I don't recall anyone asking him what it was good for!) and finding slopes of tangent lines to Fermat. It was finding the "Fundamental Theorem of Calculus", showing that these were "inverse" problems, that made Newton and Leibniz the "founders" of Calculus.

 

[arildno]

Basically, UNLESS you had the fundamental theorem of calculus, calculus would not have been of much interest.


We would then have added up all the tiny bits of stuff whenever we needed to do that.