What does "It does not have an elementary antiderivative" mean? It's a rough translation from Norwegian.
#Elementary antiderivative
lusty lodge
wet mountain
"Antiderivative" just means "integral", since derivatives and integrals are opposites.
If it does not have an elementary antiderivative, it means that the indefinite integral is not an elementary function. According to Wikipedia:
an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions (e.g., arcsin, log, or x^(1/n)).
In layman's terms (not quite accurate, but easier to understand), it means that you have to define a new function to express the indefinite integral.
lament pecan
Integrals and antiderivatives are not the same tho
wet mountain
Integrals and antiderivatives are not the same tho
Yeah. Integrals are either indefinite or definite.
lament pecan
Antiderivatives are quite literally anti-derivatives, integrals (at least in calculus II) are defined by the area under a curve... it's just a formality tho
lusty lodge
So elementary function/antiderivative refers to the indefinite/entire "figure", in contrary to the expression above (which is sufficient enough for the definite part we were looking at)?
lament pecan
Having an elementary anti-derivative just means u can "write it" down
lusty lodge
As I interpreted it, that "written" expression would span the entire graph, no?
lament pecan
Something like sin(x)/x doesn't have an elementary anti-derivative bcs u can't write down an expression for it
But the function exists
lusty lodge
Ohh, I think I understand. Was trying to get a geometric understanding of it, which proved to be abstract/difficult(?)
lament pecan
Nah, the idea is purely algebraic
lusty lodge
Yeah, saw that. Thanks for the clearance! 🙂
lament pecan
Np
lusty lodge
Also, may I ask another question regarding double integrals here?
lament pecan
Sure
lusty lodge
I'm going to find the volume of the body between {z=1-y^2 & z=x^2)
lusty lodge
Kinda uncertain what my approach should be, heh. Cause I reckon I'm supposed to derive a function and a domain to integrate for?
lament pecan
Like, just calculate one of the volumes and subtract the other one
lusty lodge
But that's the part I'm uncertain about. I tried to make expressions for the domain, defining basically a half-circle for both x & y
Oh, do it in 2 rounds? I thought I was supposed to do it between them. But I guess it would be the same, just thought they wanted it to be done that way
Not sure though
lusty lodge
Yeah, did that.
Also here’s my attempt of domain finding, where I set the equality of the z-definitions
lament pecan
Ok...
lusty lodge
(under the line)
lament pecan
Now you just do the volumes separately in respect with the places they intersect
And subtract