Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$.

(1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$.

(2) If $q=p+1$, then find the minimum value of $S$.

(3) If $pq=-1$, then  find the minimum value of $S$.
1. **Express the area \( S \) of the part enclosed by two segments \( OP, OQ \) and the parabola \( C \) in terms of \( p, q \):**

   The area \( S \) can be found by integrating the function \( y = 1 - x^2 \) from \( p \) to \( q \) and subtracting the area of the triangle formed by the points \( O, P, Q \).

   The area under the curve from \( p \) to \( q \) is given by:
   \[
   \int_p^q (1 - x^2) \, dx
   \]

   Evaluating this integral:
   \[
   \int_p^q (1 - x^2) \, dx = \left[ x - \frac{x^3}{3} \right]_p^q = \left( q - \frac{q^3}{3} \right) - \left( p - \frac{p^3}{3} \right)
   \]

   The area of the triangle \( OPQ \) can be calculated using the coordinates of \( P \) and \( Q \):
   \[
   \text{Area of } \triangle OPQ = \frac{1}{2} \left| p(1 - q^2) - q(1 - p^2) \right| = \frac{1}{2} \left| p - pq^2 - q + qp^2 \right| = \frac{1}{2} \left| p + qp^2 - q - pq^2 \right|
   \]

   Therefore, the area \( S \) is:
   \[
   S = \left( q - \frac{q^3}{3} \right) - \left( p - \frac{p^3}{3} \right) - \frac{1}{2} \left| p + qp^2 - q - pq^2 \right|
   \]

2. **If \( q = p + 1 \), then find the minimum value of \( S \):**

   Substitute \( q = p + 1 \) into the expression for \( S \):
   \[
   S(p) = \left( (p + 1) - \frac{(p + 1)^3}{3} \right) - \left( p - \frac{p^3}{3} \right) - \frac{1}{2} \left| p + (p + 1)p^2 - (p + 1) - p(p + 1)^2 \right|
   \]

   Simplify the expression:
   \[
   S(p) = \left( p + 1 - \frac{(p + 1)^3}{3} \right) - \left( p - \frac{p^3}{3} \right) - \frac{1}{2} \left| p + p^3 + p^2 - p - 1 - p^3 - 2p^2 - p \right|
   \]
   \[
   S(p) = \left( p + 1 - \frac{p^3 + 3p^2 + 3p + 1}{3} \right) - \left( p - \frac{p^3}{3} \right) - \frac{1}{2} \left| -p^2 - 1 \right|
   \]
   \[
   S(p) = \left( p + 1 - \frac{p^3}{3} - p^2 - p - \frac{1}{3} \right) - \left( p - \frac{p^3}{3} \right) - \frac{1}{2} (p^2 + 1)
   \]
   \[
   S(p) = \left( 1 - p^2 - \frac{1}{3} \right) - \frac{1}{2} (p^2 + 1)
   \]
   \[
   S(p) = \frac{2}{3} - \frac{3p^2 + 1}{6}
   \]

   To find the minimum value, take the derivative and set it to zero:
   \[
   S'(p) = -\frac{p}{2} = 0 \implies p = 0
   \]

   Substitute \( p = 0 \) back into \( S(p) \):
   \[
   S(0) = \frac{2}{3} - \frac{1}{6} = \frac{1}{2}
   \]

   Therefore, the minimum value of \( S \) is:
   \[
   \boxed{\frac{1}{2}}
   \]

3. **If \( pq = -1 \), then find the minimum value of \( S \):**

   Substitute \( q = -\frac{1}{p} \) into the expression for \( S \):
   \[
   S(p) = \left( -\frac{1}{p} - \frac{\left( -\frac{1}{p} \right)^3}{3} \right) - \left( p - \frac{p^3}{3} \right) - \frac{1}{2} \left| p + \left( -\frac{1}{p} \right)p^2 - \left( -\frac{1}{p} \right) - p \left( -\frac{1}{p} \right)^2 \right|
   \]

   Simplify the expression:
   \[
   S(p) = \left( -\frac{1}{p} + \frac{1}{3p^3} \right) - \left( p - \frac{p^3}{3} \right) - \frac{1}{2} \left| p - p - \frac{1}{p} - \frac{1}{p} \right|
   \]
   \[
   S(p) = \left( -\frac{1}{p} + \frac{1}{3p^3} \right) - \left( p - \frac{p^3}{3} \right) - \frac{1}{2} \left| -\frac{2}{p} \right|
   \]
   \[
   S(p) = \left( -\frac{1}{p} + \frac{1}{3p^3} \right) - \left( p - \frac{p^3}{3} \right) - \frac{1}{p}
   \]
   \[
   S(p) = -\frac{2}{p} + \frac{1}{3p^3} - p + \frac{p^3}{3}
   \]

   To find the minimum value, take the derivative and set it to zero:
   \[
   S'(p) = \frac{2}{p^2} - \frac{1}{p^4} - 1 + p^2 = 0
   \]

   Multiply through by \( p^4 \):
   \[
   2p^2 - 1 - p^4 + p^6 = 0
   \]

   This is a polynomial equation in \( p \). Solving this equation, we find that \( p = -1 \) is a solution.

   Substitute \( p = -1 \) back into \( S(p) \):
   \[
   S(-1) = -\frac{2}{-1} + \frac{1}{3(-1)^3} - (-1) + \frac{(-1)^3}{3}
   \]
   \[
   S(-1) = 2 - \frac{1}{3} + 1 - \frac{1}{3}
   \]
   \[
   S(-1) = 3 - \frac{2}{3} = \frac{7}{3}
   \]

   Therefore, the minimum value of \( S \) is:
   \[
   \boxed{\frac{7}{3}}
   \]