I've completely stuck with this problem:
Calculate:
$$ \int_0^{0.1} e^{-x^2} \,dx $$
solving the problem:
$$ y'(x)= e^{-x^2}, x \in [0, 0.1]$$ $$ y(0) = 0 $$
My doubt is just how to do it. I'm used to use aproximations (with Taylor, Euler, Runge-Kutta methods), but the statement always says "use X method to calculate an aproximate the value of an equation", but I have no idea about how to calculate the value of an integral.
Please, anybody could help me? Thanks in advance.
UPDATE:
The complete statement says:
i)
We can see that numerical methods which are used to solve initial value problems, are also used to calculate integrals:
Demostrate that all initial value problem of the form:
$$ y'(x)= f(t,y(t)) , t \in [t_{0}, T]$$ $$ y(t_{0}) = y_{0} $$
is equivalent to the integral equation:
$$ y(t) = y_{0} + \int_{t_0}^{T} f(s,y(s)) \,ds $$
The second section ii) is the question I ask in the beggining.
I understand the second section is an application of the first section but the integral of the problem has no anylitic solution, so that...how to solve it?
I still don't see the solution even with your help and tips..
Thanks a lot.
Hint:
$$0.1-\frac{0.1^3}{3}+\frac{0.1^5}{5\cdot2!}-\frac{0.1^7}{7\cdot3!}+\cdots$$