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The second part of the Fundamental Theorem of Calculus is given as, $$\int_{a(x)}^{b(x)}f(t)dt=F(b(x)) - F(a(x))$$ Where $F$ is the antiderivative of $f$.

Now, let us consider the function $f(t) = e^{t^2}$

It is evident that there is no elementary antiderivative of $f(t)$. Does the question of computing the area under $f(t)$, the area mathematically given by evaluating $\int_{a(x)}^{b(x)}e^{t^2}dt$, go unanswered while trying to implement the second part of the theorem?

R004
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  • You can't express $\int e^{t^2} dt$ (ie the antiderivate of $e^{t^2}$) in terms of elementary functions. – skyking Jun 14 '17 at 09:18
  • That is known. It is also known that the area under the curve exists. I am interested to know how we can go about evaluating the area. Is calculating the limits of Riemann sums one of the ways? – R004 Jun 14 '17 at 09:23
  • I think you should elaborate exactly what you're trying to achieve here, this looks like an X-Y problem (https://meta.stackexchange.com/questions/66377/what-is-the-xy-problem). It's not necessarily the case that the FToC is the tool of choice for your problem. – skyking Jun 14 '17 at 09:28
  • It seems you are asking how to numerically calculate a definite integral. That is a broad topic with many answers, thus off-topic for this site. To do research, search for "numeric integration" and "quadrature." – Rory Daulton Jun 14 '17 at 10:35
  • Related: https://math.stackexchange.com/questions/2226732/proof-that-the-area-under-a-curve-is-the-definite-integral-without-the-fundamen/2226894#2226894 – Ethan Bolker Jun 14 '17 at 14:48

1 Answers1

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An antiderivative exists, just not in terms of elementary functions. Put $F(x):=\int_0^xe^{t^2}dt$. This can be evaluated numerically for each $x\in\mathbb R$ using, e.g. Riemann sums. Then $F(b)-F(a)=\int_a^be^{t^2}dt$, and similarly with $a=a(x)$, $b=b(x)$.

user254433
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  • I was speculating that evaluating the limits of Riemann sums is the way( or is one of the ways ). – R004 Jun 14 '17 at 09:25
  • I, however, am yet to understand the exponential series( which kicks in later during the course ). My calculator evaluates this sum. It takes quite some time, though. I wonder how my calculator evaluates the area in this particular case. – R004 Jun 14 '17 at 09:27
  • Yes, Riemann sums could be used, although trapezoidal or Simpson's rules are to be preferred since they are more accurate. Using the Taylor series for the exponential would be atypical, for the reason you mentioned. – user254433 Jun 14 '17 at 09:30
  • @R004 Probably using a numerical method for integration (for example the Simpson's rule). Series is not always the method of choice as this function would diverge quite fast making a polynomial a bad approximation for larger $x$. – skyking Jun 14 '17 at 09:31
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    @R004 For the actual exponential series are being used, but the calculator also uses exponential laws to bring the effective argument near $0$ (that technique can't be used for evaluating the antiderivate to $e^{x^2}$ AFAIK). – skyking Jun 14 '17 at 09:33