restart;
# Quiz 9 Question 1:
f:=t->t*sin(t);
g:=t->t*cos(t);


plot([f(t),g(t),t=0..8*Pi],numpoints=5000);

L1:=int(sqrt(D(f)(t)^2+D(g)(t)^2),t=0..8*Pi);

evalf(L1);

# Question 2:
u:=t->sqrt(t)*sin(t);
v:=t->sqrt(t)*cos(t);


plot([u(t),v(t),t=0..8*Pi],numpoints=5000);

L2:=int(sqrt(D(u)(t)^2+D(v)(t)^2),t=0..8*Pi);

L3:=Int(sqrt(D(u)(t)^2+D(v)(t)^2),t=0..8*Pi);

evalf(L3);

# Question 3
with(IntegrationTools);

I1:=Int(1/sqrt(1-x^2),x=0..1/2);

Change(x=sin(theta),I1);

# Question 4
T1:=Sum(x^k/k!*Product(alpha-j,j=0..k-1),k=0..n);

T2:=subs({x=-1/2,alpha=-1/2},T1);

S:=unapply(T2,n);

T3:=seq(value(S(n)),n=1..5);

evalf(T3);