In my calc class, we've been on integration for a while, and all the time that we have been learning more advanced integration techniques, I'm still dumbfounded on why finding and antiderivative will yield the area under a curve. I easily understand how to find an antiderivative, and understand why F(a) - F(b) gives the area between the two points, but am absolutely lost when it comes to why the antiderivative of f (F) will yield area when the points are plugged in. I wouldn't waste anyone here's time if I understood the proofs I've seen, trust me. Could someone weigh in with a simple enough proof, or just help me understand the concept? Thanks!
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1$\begingroup$ Look at :math.stackexchange.com/questions/1189262/…. There is a figure in my answer that can be useful. $\endgroup$– Emilio NovatiMar 22, 2015 at 20:39
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$\begingroup$ Self-plug: something I wrote for my students on this topic. An Intuitive Introduction to Calculus $\endgroup$– davidlowryduda ♦Mar 22, 2015 at 21:02
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1 Answer
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Think of the interval [a,b]
So you know the f(a) is the value of the function at $x=a$ and f(b) is the value of the function at $x=b$. Think of this as drawing a really thin line from the x-axis to the value of the function.
In order to get the area under the curve we need to draw infinitely thin lines at all the points on the interval [a,b]. That is we need to draw infinitely thin lines from the x-axis to the functions value over the interval [a,b]. We will define this as f(F). This will eventually span a shape under the graph from the x axis over [a,b] to the points f(x) $a\leqslant x \leqslant b$
