2014年山东ACM第五届省赛 angry_birds_again_and_again（数学积分）

angry_birds_again_and_again

Time Limit: 2000ms Memory limit: 65536K 有疑问？点这里^_^

题目描述

The problems called "Angry Birds" and "Angry Birds Again and Again" has been solved by many teams in the series of contest in 2011 Multi-University Training Contest.

This time we focus on the yellow bird called Chuck. Chuck can pick up speed and distance when tapped.

You can assume that before tapped, Chuck flies along the parabola. When tapped, it changes to fly along the tangent line. The Chuck starts at the coordinates (0, 0). Now you are given the coordinates of the pig (Px, 0), the x-coordinate of the tapping
position (Tx) and the initial flying angle of Chuck (α).



∠AOx = α
Please calculate the area surrounded by Chuck’s path and the ground.(The area surrounded by the solid line O-Tapping position-Pig-O)

输入

The first line contains only one integer T (T is about 1000) indicates the number of test cases. For each case there are two integers, px tx, and a float number α.（0 < Tx ≤ Px ≤ 1000, 0 < α < 

）
.

输出

One line for each case specifying the distance rounded to three digits.

示例输入

1
2 1 1.0

示例输出

0.692

解：看到题目很多人都会往物理方向思考，实际上这种想法是错误的。大致题意就是给出图中点TX的横坐标tx，点Pig的横坐标px，角A，求图中实线部分的面积。就是一部分弧面的面积和三角形的面积。

设方程 y=ax^2+bx+c ，图中曲线经过原点，所以c=0.

对方程求导 y'=2ax+b , 此时 y'代表斜率，将原点(0,0)带入得：y'=b即该点的斜率，所以b=tan( A）

然后根据两条切线的斜率求出a的值a=(-b*tx-b*(px-tx))/(2*tx*(px-tx)+tx*tx)。

这样对y进行0到tx积分求出弧面的面积，再加上三角形的面积。

s=((a*tx*tx*tx)*(1/3.0)+(b*tx*tx)*0.5)+((a*tx*tx+b*tx)*(px-tx)*0.5)。注：double型要除以3.0而不能是3。

#include<stdio.h>
#include<math.h>
int main()
{
int n;
scanf("%d",&n);
while(n--)
{
double px,tx,A;
double s;
scanf("%lf %lf %lf",&px,&tx,&A);
double b=tan(A);
double a=(-b*tx-b*(px-tx))/(2*tx*(px-tx)+tx*tx);
s=((a*tx*tx*tx)*(1/3.0)+(b*tx*tx)*0.5)+((a*tx*tx+b*tx)*(px-tx)*0.5);
printf("%.3lf\n",s);
}
}