> # Askisi 1
> f:=b*sqrt(1-x^2/a^2);
                                     (1/2)
                             /     2\     
                             |    x |     
                      f := b |1 - --|     
                             |     2|     
                             \    a /     
> E:=2*int(f, x=-a..a);
                                b Pi   
                         E := ---------
                                  (1/2)
                              /1 \     
                              |--|     
                              | 2|     
                              \a /     
> # Askisi 2
> 
> restart:
> f:=3*sqrt(1-x^2/4);
                                     (1/2)
                           3 /     2\     
                      f := - \4 - x /     
                           2              
> g:=2*sqrt(1-x^2/9);
                                     (1/2)
                           2 /     2\     
                      g := - \9 - x /     
                           3              
> plot({f,g},x=-4..4);

> sols:=solve(f=g,x);
                          6    (1/2)  6    (1/2)
                sols := - -- 13     , -- 13     
                          13          13        
> E1:=int(f,x=-2..sols[1]);
                   3      18           /3    (1/2)\
             E1 := - Pi - -- - 3 arcsin|-- 13     |
                   2      13           \13        /
> E2:=int(g, x=sols[1]..sols[2]);
                      36           /2    (1/2)\
                E2 := -- + 6 arcsin|-- 13     |
                      13           \13        /
> E3:=int(f, x=sols[2]..2);
                   3      18           /3    (1/2)\
             E3 := - Pi - -- - 3 arcsin|-- 13     |
                   2      13           \13        /
> E:=2*(E1+E2+E3);
                        /3    (1/2)\            /2    (1/2)\
   E := 6 Pi - 12 arcsin|-- 13     | + 12 arcsin|-- 13     |
                        \13        /            \13        /
> evalf(E);
                          14.11206248
> 
> # Askisi 3 
> 
> restart:
> f:=sqrt(x^2+3*x)-sqrt(x^2-x);
                             (1/2)           (1/2)
                   / 2      \        / 2    \     
              f := \x  + 3 x/      - \x  - x/     
> limit(f,x=infinity);
                               2
> g:= x*(exp(1/x)-1);
                             /   /1\    \
                      g := x |exp|-| - 1|
                             \   \x/    /
> limit(g, x=infinity);
                               1
> limit(g, x=-infinity);
                               1
> limit(g, x=0);
                               g
> # To orio den uparxei. E3etazw ta pleurika oria
> limit(g,x=0,left);
                               0
> limit(g,x=0,right);
                            infinity
> h:=int(1/log(t),t=2..x+1/x);
                        /  1                 1\
                h := int|-----, t = 2 .. x + -|
                        \ln(t)               x/
> limit(h, x=infinity);
                            infinity
> limit(h, x=0, right);
                            infinity
> 
> # Askisi 4
> 
> restart:
> f:=sqrt(x);
                                (1/2)
                          f := x     
> T:=taylor(f, x=1,2);
                          1          /       2\
                 T := 1 + - x - 1 + O\(x - 1) /
                          2                    
> P:=convert(T,polynom);
                               1   1  
                          P := - + - x
                               2   2  
> plot({f,P},x=0..2);

> # Parathrw oti to grammiko meros tou anaptugmatos Taylor ths f gurw apo to x0 einai h e3iswsh ths
> # efapromenhs sto shmeio (x0,f(x0)).
> 
> efaptomeni := proc(g,a)
> local T,P:
> T:=taylor(f,x=a,2):
> P:=convert(T,polynom):
> RETURN(P)
> end proc:
> efaptomeni(f,1);
                            1   1  
                            - + - x
                            2   2  
>