Curvy Little Bottles

On her bike rides around Warsaw, Jill came across a shop selling interesting glass bottles. She thought it would be a fun project to use such bottles for measuring liquids, but that would require placing marks on a bottle to indicate various volumes. Where should those volume marks go?

Jill formalized the problem as follows. A bottle is formed by revolving, around the $x$-axis, the region under the graph of a polynomial $P$ between $x = x_{low}$ and $x = x_{high}$. Thus the $x$-axis runs vertically through the center of the bottle. The bottom of the bottle is a solid circular disk at $x = x_{low}$, and the top of the bottle, at $x = x_{high}$, is left open. The value of $P$ is greater than zero everywhere between $x_{low}$ and $x_{high}$.

The volume of the bottle between $x_{low}$ and a height $x$ is $\int_{x_{low}}^{x} \pi\,P(t)^2\,dt$. For example, the bottle formed by $P(x) = 4 - 0.25x$ with $x_{low} = 0$ and $x_{high} = 12$ has a bottom circle of radius $4$, a top opening of radius $1$, and height $12$.

Given a polynomial $P$, the bounds $x_{low}$ and $x_{high}$, and the volume increment between successive marks, compute the distances up from $x_{low}$ of the marks at successive volume increments. A mark cannot be placed past the top of the bottle, and no more than the first $8$ increments should be marked.

The input contains several test cases, one after another, until end of file. Each test case consists of three lines of bottle data:

• Line 1: $n$, the degree of the polynomial, an integer with $0 \le n \le 10$.
• Line 2: $a_0, a_1, \ldots, a_n$, the real coefficients of the polynomial $P$, where $a_0$ is the constant term and $a_i$ is the coefficient of $x^i$. For each $i$, $-100 \le a_i \le 100$, and $a_n \ne 0$.
• Line 3: two real values $x_{low}$ and $x_{high}$, the boundaries of the bottle, with $-100 \le x_{low} < x_{high} \le 100$ and $x_{high} - x_{low} > 0.1$; followed by $inc$, an integer volume increment between successive marks, with $1 \le inc \le 500$.

For each test case, print two lines. On the first line, print the case number and the volume of the full bottle, in the form Case k: V. On the second line, print the increasing sequence of at most $8$ successive distances up from the bottom of the bottle for the volume marks, separated by single spaces. All volumes and distances must be accurate to two decimal places. If the bottle does not have enough volume for even one mark, print insufficient volume on the second line instead.

It is guaranteed that no mark falls within $0.01$ of the top of the bottle, that the volume of the bottle does not exceed $1000$, and that all rounded mark distances on a bottle differ by at least $0.05$.